Mathematics for the International Students: IB Dipolma HL Core

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HAESE

&

HARRIS PUBLICATIONS

Specialists in mathematics publishing

Mathematics

for the international student

Mathematics HL (Core) second edition

Paul Urban David Martin Robert Haese Sandra Haese Michael Haese Mark Humphries

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for use with IB Diploma Programme black

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MATHEMATICS FOR THE INTERNATIONAL STUDENT Mathematics HL (Core) second edition Paul Urban David Martin Robert Haese Sandra Haese Michael Haese Mark Humphries

B.Sc.(Hons.),B.Ec. B.A.,B.Sc.,M.A.,M.Ed.Admin. B.Sc. B.Sc. B.Sc.(Hons.),Ph.D. B.Sc.(Hons.)

Haese & Harris Publications 3 Frank Collopy Court, Adelaide Airport, SA 5950, AUSTRALIA Telephone: +61 8 8355 9444, Fax: + 61 8 8355 9471 Email: [email protected] www.haeseandharris.com.au Web: National Library of Australia Card Number & ISBN 978-1-876543-11-2 © Haese & Harris Publications 2008 Published by Raksar Nominees Pty Ltd. 3 Frank Collopy Court, Adelaide Airport, SA 5950, AUSTRALIA First Edition Reprinted Second Edition

2004 2005 three times (with minor corrections), 2006, 2007 2008

Cartoon artwork by John Martin. Artwork by Piotr Poturaj and David Purton. Cover design by Piotr Poturaj. Computer software by David Purton, Thomas Jansson and Troy Cruickshank. Typeset in Australia by Susan Haese (Raksar Nominees). Typeset in Times Roman 10\Qw_ /11\Qw_\\"

The textbook and its accompanying CD have been developed independently of the International Baccalaureate Organization (IBO). The textbook and CD are in no way connected with, or endorsed by, the IBO. This book is copyright. Except as permitted by the Copyright Act (any fair dealing for the purposes of private study, research, criticism or review), no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher. Enquiries to be made to Haese & Harris Publications. Copying for educational purposes: Where copies of part or the whole of the book are made under Part VB of the Copyright Act, the law requires that the educational institution or the body that administers it has given a remuneration notice to Copyright Agency Limited (CAL). For information, contact the Copyright Agency Limited. Acknowledgements: While every attempt has been made to trace and acknowledge copyright, the authors and publishers apologise for any accidental infringement where copyright has proved untraceable. They would be pleased to come to a suitable agreement with the rightful owner.

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Disclaimer: All the internet addresses (URL’s) given in this book were valid at the time of printing. While the authors and publisher regret any inconvenience that changes of address may cause readers, no responsibility for any such changes can be accepted by either the authors or the publisher.

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FOREWORD Mathematics for the International Student: Mathematics HL has been written to reflect the syllabus for the two-year IB Diploma Mathematics HL course. It is not our intention to define the course. Teachers are encouraged to use other resources. We have developed the book independently of the International Baccalaureate Organization (IBO) in consultation with many experienced teachers of IB Mathematics. The text is not endorsed by the IBO. This second edition builds on the strengths of the first edition. Many excellent suggestions were received from teachers around the world and these are reflected in the changes. In some cases sections have been consolidated to allow for greater efficiency. Changes have also been made in response to the introduction of a calculator-free examination paper. A large number of questions, including some to challenge even the best students, have been added. In particular, the final chapter contains over 200 miscellaneous questions, some of which require the use of a graphics calculator. These questions have been included to provide more difficult challenges for students and to give them experience at working with problems that may or may not require the use of a graphics calculator. The combination of textbook and interactive Student CD will foster the mathematical development of students in a stimulating way. Frequent use of the interactive features on the CD is certain to nurture a much deeper understanding and appreciation of mathematical concepts. The book contains many problems from the basic to the advanced, to cater for a wide range of student abilities and interests. While some of the exercises are simply designed to build skills, every effort has been made to contextualise problems, so that students can see everyday uses and practical applications of the mathematics they are studying, and appreciate the universality of mathematics. Emphasis is placed on the gradual development of concepts with appropriate worked examples, but we have also provided extension material for those who wish to go beyond the scope of the syllabus. Some proofs have been included for completeness and interest although they will not be examined. For students who may not have a good understanding of the necessary background knowledge for this course, we have provided printable pages of information, examples, exercises and answers on the Student CD. To access these pages, simply click on the ‘Background knowledge’ icons when running the CD. It is not our intention that each chapter be worked through in full. Time constraints will not allow for this. Teachers must select exercises carefully, according to the abilities and prior knowledge of their students, to make the most efficient use of time and give as thorough coverage of work as possible. Investigations throughout the book will add to the discovery aspect of the course and enhance student understanding and learning. Many Investigations could be developed into portfolio assignments. Teachers should follow the guidelines for portfolio assignments to ensure they set acceptable portfolio pieces for their students that meet the requirement criteria for the portfolios. Review sets appear at the end of each chapter and a suggested order for teaching the two-year course is given at the end of this Foreword. The extensive use of graphics calculators and computer packages throughout the book enables students to realise the importance, application and appropriate use of technology. No single aspect of technology has been favoured. It is as important that students work with a pen and paper as it is that they use their calculator or graphics calculator, or use a spreadsheet or graphing package on computer.

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The interactive features of the CD allow immediate access to our own specially designed geometry packages, graphing packages and more. Teachers are provided with a quick and easy way to demonstrate concepts, and students can discover for themselves and re-visit when necessary.

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Instructions appropriate to each graphic calculator problem are on the CD and can be printed for students. These instructions are written for Texas Instruments and Casio calculators. In this changing world of mathematics education, we believe that the contextual approach shown in this book, with the associated use of technology, will enhance the students’ understanding, knowledge and appreciation of mathematics, and its universal application. We welcome your feedback. Email: [email protected] Web: www.haeseandharris.com.au

PMU DCM RCH SHH PMH MAH

ACKNOWLEDGEMENTS The authors and publishers would like to thank all those teachers who have offered advice and encouragement. Many of them have read page proofs and made constructive comments and suggestions. Particular thanks go to Stephen Hobbs who has given generously of his time in reviewing the first edition and making suggestions for improvement in this second edition. Thanks are also due to Dr. Andrzej Cichy, Peter Blythe, Brendan Watson, Myrricia Holmann, Jeff Jones, Mark Willis, John Poole and Marjut Mäenpää. We acknowledge the contributions of John Owen and Mark Bruce in the preparation of the first edition and we also want to thank others who provided assistance – they include: Cameron Hall, Fran O'Connor, Glenn Smith, Anne Walker, Malcolm Coad, Ian Hilditch, Phil Moore, Julie Wilson, Kerrie Clements, Margie Karbassioun, Brian Johnson, Carolyn Farr, Rupert de Smidt, Terry Swain, Marie-Therese Filippi, Nigel Wheeler, Sarah Locke, Rema George. The publishers wish to make it clear that acknowledging these individual does not imply any endorsement of this book by any of them and all responsibility for content rests with the authors and publishers.

TEACHING THE TWO-YEAR COURSE – A SUGGESTED ORDER Teachers are encouraged to carefully check the BACKGROUND KNOWLEDGE sections supplied on the accompanying CD to ensure that basics have been mastered relatively early in the two-year HL course. Some of these topics naturally occur at the beginning of a specific chapter, as indicated in the table of contents. Click on the BACKGROUND KNOWLEDGE active icons to access the printable pages on the CD. Teachers will have their personal preferences for the order in which the chapters are tackled. A suggestion is to work progressively from Chapter 1 through to Chapter 20, but leave Chapters 9, 15 and, possibly, 16 for the second year. The remaining chapters can be worked through in order. Alternatively, for the first year, students could work progressively from Chapter 1 to Chapter 23 but not necessarily including chapters 7, 15 and 16. Chapter 9 ‘Mathematical Induction’ could also be attempted later, perhaps early in the second year. In some parts of the world, the topics of Polynomials, Complex Numbers, 3-D Vector Geometry and Calculus are not usually covered until the final year of school. Another approach could be to teach just those topics that are included in the Mathematics SL syllabus in the first year and leave the remaining topics for completion in the second year.

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However, it is acknowledged that there is no single best way for all teachers to work through the syllabus. Individual teachers have to consider particular needs of their students and other requirements and preferences that they may have.

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USING THE INTERACTIVE STUDENT CD The CD is ideal for independent study. Frequent use will nurture a deeper understanding of Mathematics. Students can revisit concepts taught in class and undertake their own revision and practice. The CD also has the text of the book, allowing students to leave the textbook at school and keep the CD at home. The icon denotes an Interactive Link on the CD. Simply ‘click’ the icon to access a range of interactive features: w w w w w w

spreadsheets video clips graphing and geometry software graphics calculator instructions computer demonstrations and simulations background knowledge (as printable pages)

INTERACTIVE LINK

For those who want to make sure they have the prerequisite levels of understanding for this course, printable pages of background information, examples, exercises and answers are provided on the CD. Click the ‘Background knowledge’ icon on pages 12 and 248. Graphics calculators: Instructions for using graphics calculators are also given on the CD and can be printed. Instructions are given for Texas Instruments and Casio calculators. Click on the relevant icon (TI or C) to access printable instructions. Examples in the textbook are not always given for both types of calculator. Where that occurs, click on the relevant icon to access the instructions for the other type of calculator.

TI C

NOTE ON ACCURACY Students are reminded that in assessment tasks, including examination papers, unless otherwise stated in the question, all numerical answers must be given exactly or to three significant figures.

HL & SL COMBINED CLASSES Refer to our website www.haeseandharris.com.au for guidance in using this textbook in HL and SL combined classes.

HL OPTIONS This is a companion to the Mathematics HL (Core) textbook. It offers coverage of each of the following options: w w w w

Topic 8 –Statistics and probability Topic 9 – Sets, relations and groups Topic 10 – Series and differential equations Topic 11 – Discrete mathematics

In addition, coverage of the Geometry option for students undertaking the IB Diploma course Further Mathematics is presented on the CD that accompanies the HL Options book.

SUPPLEMENTARY BOOKS

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A separate book of WORKED SOLUTIONS gives the fully worked solutions for every question (discussions, investigations and projects excepted) in each chapter of the Mathematics HL (Core) textbook. The HL (CORE) EXAMINATION PREPARATION & PRACTICE GUIDE offers additional questions and practice exams to help students prepare for the Mathematics HL examination. For more information email [email protected]

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6

TABLE OF CONTENTS

TABLE OF CONTENTS SYMBOLS AND NOTATION USED IN THIS BOOK

10

BACKGROUND KNOWLEDGE

12

F G H

Graphs of exponential functions Growth and decay The natural exponential ‘e’ Review set 3A Review set 3B Review set 3C

4

LOGARITHMS

A B C D E F G H

Logarithms Logarithms in base 10 Laws of logarithms Natural logarithms Exponential equations using logarithms The change of base rule Graphs of logarithmic functions Growth and decay Review set 4A Review set 4B Review set 4C Review set 4D

5

GRAPHING AND TRANSFORMING FUNCTIONS 125

A B C D

Families of functions Transformations of graphs Simple rational functions Further graphical transformations Review set 5A Review set 5B

6

QUADRATIC EQUATIONS AND FUNCTIONS

143

A B C D E F G H

Solving quadratic equations (Review) The discriminant of a quadratic The sum and product of the roots Graphing quadratic functions Finding a quadratic from its graph Where functions meet Problem solving with quadratics Quadratic optimisation Review set 6A Review set 6B Review set 6C Review set 6D Review set 6E

145 149 152 153 161 165 167 170 173 174 175 175 176

to access, ‘click’ active icon on CD

Solutions of real quadratics with Δ < 0 Complex numbers Real polynomials Roots, zeros and factors

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A B C D

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78 79 80 84 87

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Index notation Evaluating powers Index laws Algebraic expansion and factorisation Exponential equations

75

A B C D E

COMPLEX NUMBERS AND POLYNOMIALS

25

7

42 44 46 49 50 51

5

77

50

EXPONENTIALS

25

3

0

54 54 56 59 65 72 74 75 76

5

53

Number patterns Sequences of numbers Arithmetic sequences Geometric sequences Series Miscellaneous problems Review set 2A Review set 2B Review set 2C

95

SEQUENCES AND SERIES

A B C D E F

100

2

H I J

50

18 21 27 28 32 35 41

75

17

Relations and functions Function notation, domain and range Composite functions, f ± g Sign diagrams Inequalities (inequations) The modulus function 1 The reciprocal function x x Asymptotes of other rational functions Inverse functions Functions which have inverses Review set 1A Review set 1B Review set 1C

25

FUNCTIONS

A B C D E F G

0

1

5

CD CD CD CD CD CD CD CD CD CD CD CD CD

95

Surds and radicals Scientific notation (Standard form) Number systems and set notation Algebraic simplification Linear equations and inequalities Modulus or absolute value Product expansion Factorisation Formula rearrangement Adding and subtracting algebraic fractions Congruence and similarity Coordinate geometry ANSWERS

100

A B C D E F G H I J K L

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88 91 95 98 99 99

101 102 104 106 110 112 114 115 120 122 123 123 124

126 128 133 137 140 141

177 178 180 188 193

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Graphing polynomials Theorems for real polynomials Review set 7A Review set 7B Review set 7C

201 208 210 211 212

8

COUNTING AND THE BINOMIAL EXPANSION

213

A B C D E F G

The product principle Counting paths Factorial notation Permutations Combinations Binomial expansions The general binomial expansion Review set 8A Review set 8B

9

MATHEMATICAL INDUCTION

A B C

The process of induction The principle of mathematical induction Indirect proof (extension) Review set 9A Review set 9B Review set 9C

G H I J K L M

214 216 217 219 223 226 229 231 232

A B C D E F G

234 236 244 245 245 246

D

A B C D E F G H I J

CD 248 250 253 263 266 267 268

11 NON-RIGHT ANGLED TRIANGLE TRIGONOMETRY 269 270 272 277 280 281

12 ADVANCED TRIGONOMETRY

283

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Vectors Operations with vectors 2-D vectors in component form 3-D coordinate geometry 3-D vectors in component form Algebraic operations with vectors Parallelism Unit vectors The scalar product of two vectors The vector product of two vectors Review set 14A Review set 14B Review set 14C Review set 14D Review set 14E

Complex numbers as 2-D vectors Modulus, argument, polar form De Moivre's Theorem Roots of complex numbers Further complex number problems Review set 15A Review set 15B Review set 15C

16 LINES AND PLANES IN SPACE A B

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285 288 293 296 297 299

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Observing periodic behaviour The sine function Modelling using sine functions The cosine function The tangent function Trigonometric equations

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A B C D E F

Matrix structure Matrix operations and definitions The inverse of a 2 × 2 matrix 3 × 3 and larger matrices Solving systems of linear equations Solving systems using row operations Induction with matrices Review set 13A Review set 13B Review set 13C Review set 13D Review set 13E

15 COMPLEX NUMBERS

95

The cosine rule The sine rule Using the sine and cosine rules Review set 11A Review set 11B

5

A B C

305 307 309 310 314 318 318 319 320 321 322

14 VECTORS IN 2 AND 3 DIMENSIONS

10 THE UNIT CIRCLE AND RADIAN MEASURE 247

A B C

Using trigonometric models Reciprocal trigonometric functions Trigonometric relationships Compound angle formulae Double angle formulae Trigonometric equations in quadratic form Trigonometric series and products Review set 12A Review set 12B Review set 12C Review set 12D

13 MATRICES

233

BACKGROUND KNOWLEDGE – TRIGONOMETRY WITH RIGHT ANGLED TRIANGLES – Printable pages Radian measure Arc length and sector area The unit circle and the basic trigonometric ratios Areas of triangles Review set 10A Review set 10B Review set 10C

7

Lines in 2-D and 3-D Applications of a line in a plane

100

E F

TABLE OF CONTENTS

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371 372 375 383 388 390 393 398 400 402 407 416 417 418 419 420

421 422 425 438 441 445 445 446 447

449 451 456

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461 466 471 473 477 478 479 481

17 DESCRIPTIVE STATISTICS A B C D E F G

L

A B C D E F G H

483

Continuous numerical data and histograms Measuring the centre of data Cumulative data Measuring the spread of data Statistics using technology Variance and standard deviation The significance of standard deviation Review set 17A Review set 17B

18 PROBABILITY A B C D E F G H I J K

21 APPLICATIONS OF DIFFERENTIAL CALCULUS

485 489 500 502 510 512 518 520 522

528 532 533 537 541 543 546 549 554 558

668 673 677 679 683 684 685

23 DERIVATIVES OF CIRCULAR FUNCTIONS AND RELATED RATES

687

A B

560 562 564 565 566 568

Derivatives of circular functions The derivatives of reciprocal circular functions The derivatives of inverse circular functions Maxima and minima with trigonometry Related rates Review set 23A Review set 23B

C D E

20 DIFFERENTIAL CALCULUS

589

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Antidifferentiation The fundamental theorem of calculus Integration Integrating eax+b and (ax + b)n Integrating f (u)u0 (x) by substitution Integrating circular functions Definite integrals Review set 24A Review set 24B Review set 24C

25 APPLICATIONS OF INTEGRATION A B

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592 595 600 604 607 611 616 618 619 620

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The derivative function Derivatives at a given x-value Simple rules of differentiation The chain rule Product and quotient rules Tangents and normals Higher derivatives Review set 20A Review set 20B Review set 20C

100

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A B C D E F G

24 INTEGRATION

Finding areas between curves Motion problems

25

570 574 577 579 586

5

Limits Finding asymptotes using limits Trigonometric limits Calculation of areas under curves Review set 19

622 623 627 634 642 647 652 661 664 665 666

Exponential e Natural logarithms Derivatives of logarithmic functions Applications Some special exponential functions Review set 22A Review set 22B

A B C D E

19 INTRODUCTION TO CALCULUS 569 A B C D

621

22 DERIVATIVES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS 667

525

Experimental probability Sample space Theoretical probability Compound events Using tree diagrams Sampling with and without replacement Binomial probabilities Sets and Venn diagrams Laws of probability Independent events Probabilities using permutations and combinations Bayes’ theorem Review set 18A Review set 18B Review set 18C Review set 18D

Time rate of change General rates of change Motion in a straight line Some curve properties Rational functions Inflections and shape Optimisation Implicit differentiation Review set 21A Review set 21B Review set 21C

95

Relationship between lines Planes and distances Angles in space The intersection of two or more planes Review set 16A Review set 16B Review set 16C Review set 16D

100

C D E F

50

TABLE OF CONTENTS

75

8

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688 693 694 697 699 704 705

707 708 710 715 720 722 724 730 734 735 736

737 738 744

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Problem solving by integration Review set 25A Review set 25B Review set 25C

748 752 753 755

26 VOLUMES OF REVOLUTION

757

Solids of revolution Volumes for two defining functions Review set 26

758 762 765

27 FURTHER INTEGRATION AND DIFFERENTIAL EQUATIONS

28 STATISTICAL DISTRIBUTIONS OF DISCRETE RANDOM VARIABLES

786 788 791 794 801 807 810 812

29 STATISTICAL DISTRIBUTIONS OF CONTINUOUS RANDOM VARIABLES A B C D

821 828 830 831

833

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INDEX

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ANSWERS

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30 MISCELLANEOUS QUESTIONS

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Continuous probability density functions Normal distributions The standard normal distribution (Z-distribution) Applications of the normal distribution Review set 29A Review set 29B

95

E F

785

Discrete random variables Discrete probability distributions Expectation The measures of a discrete random variable The binomial distribution The Poisson distribution Review set 28A Review set 28B

100

A B C D

769 771 773 774 783 784

50

B C D E

768

5

A

767

1 1 The integrals of p 2 and 2 x + a2 a ¡ x2 Further integration by substitution Integration by parts Miscellaneous integration Separable differential equations Review set 27A Review set 27B

75

A B

25

C

9

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SYMBOLS AND NOTATION USED IN THIS BOOK

> or >

N

the set of positive integers and zero, f0, 1, 2, 3, ......g

Z

the set of integers, f0, §1, §2, §3, ......g

Z+

the set of positive integers, f1, 2, 3, ......g

Q

R +

R

[a, b]

the set of real numbers

] a, b [

i z

a complex number

z¤

the complex conjugate of z

jzj

the modulus of z

arg z

the argument of z

d

the common difference of an arithmetic sequence

r

the common ratio of a geometric sequence

Sn

the sum of the first n terms of a sequence, u1 + u2 + ::::: + un

S1 or S ui

n r

the set with elements x1 , x2 , ..... the set of all x such that

f is a function under which each element of set A has an image in set B

f : x 7! y

f is a function under which x is mapped to y the image of x under the function f

2 =

is not an element of

f ¡1

the inverse function of the function f

?

the empty (null) set

f ±g

the composite function of f and g

lim f (x)

the limit of f (x) as x tends to a

U

the universal set

[

union

\

intersection

µ

is a subset of

A0 p a , na

x!a

dy dx f 0 (x)

a2 ,

d2 y

the complement of the set A a to the power of (if a > 0 then

p a

a to the power (if a > 0 then

jxj

1 , n

p n

1 , 2

p

dx2

nth root of a

f 00 (x)

a > 0)

dn y dxn

square root of a a > 0)

f (n) (x)

R

the n modulus or absolute value of x, that is x for x > 0 x2R ¡x for x < 0 x 2 R

´

Z

y dx y dx

identity or is equivalent to

¼

is approximately equal to

>

is greater than

magenta

a

ex

yellow

the derivative of f (x) with respect to x the second derivative of y with respect to x the second derivative of f (x) with respect to x the nth derivative of y with respect to x the nth deriviative of f (x) with respect to x the indefinite integral of y with respect to x

95

the definite integral of y with respect to x between the limits x = a and x = b exponential function of x

100

50

75

25

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the derivative of y with respect to x

b

loga x

25

n! r!(n ¡ r)!

f (x)

1 n

0

u1 + u2 + ::::: + un

is an element of

2

5

the sum to infinity of a sequence, u1 + u2 + :::::

f : A!B

the number of elements in the finite set A

fx j ......

the open interval a < x < b the nth term of a sequence or series

³ ´

the real part of z the imaginary part of z

n(A)

the closed interval a 6 x 6 b

i=1

Re z

1

un

n X

Im z fx1 , x2 , ....g

is not less than

the set of positive rational numbers, fx j x > 0 , x 2 Q g

the set of complex numbers, fa + bi j a, b 2 R g p ¡1

is less than or equal to is not greater than

¥

the set of positive real numbers, fx j x > 0, x 2 R g

C

is less than

· or 6

the set of rational numbers

+

Q

0, y > 0

(2)

y These 13 points form a relation.

x

x

FUNCTIONS A function, sometimes called a mapping, is a relation in which no two different ordered pairs have the same x-coordinate (first member). We can see from the above definition that a function is a special type of relation.

TESTING FOR FUNCTIONS Algebraic Test: If a relation is given as an equation, and the substitution of any value for x results in one and only one value of y, we have a function. For example: ² y = 3x ¡ 1 is a function, as for any value of x there is only one value of y ² x = y 2 is not a function since if x = 4, say, then y = §2. Geometric Test or “Vertical Line Test”:

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If we draw all possible vertical lines on the graph of a relation, the relation: ² is a function if each line cuts the graph no more than once ² is not a function if at least one line cuts the graph more than once.

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20

FUNCTIONS (Chapter 1)

Example 1 Which of the following relations are functions? a b y

c

y

y

x

x

x

a

b

y

c

y

y

x

x

a function

not a function

x a function

GRAPHICAL NOTE ²

If a graph contains a small open circle such as

²

If a graph contains a small filled-in circle such as

²

If a graph contains an arrow head at an end such as then the graph continues indefinitely in that general direction, or the shape may repeat as it has done previously.

, this point is not included. , this point is included.

EXERCISE 1A 1 Which of the following sets of ordered pairs are functions? Give reasons. a f(1, 3), (2, 4), (3, 5), (4, 6)g b f(1, 3), (3, 2), (1, 7), (¡1, 4)g c f(2, ¡1), (2, 0), (2, 3), (2, 11)g d f(7, 6), (5, 6), (3, 6), (¡4, 6)g e f(0, 0), (1, 0), (3, 0), (5, 0)g f f(0, 0), (0, ¡2), (0, 2), (0, 4)g 2 Use the vertical line test to determine which of the following relations are functions: a b c d y

y

y

x

e

y x

x

f

g

y

x

h

y

y

y

x x

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3 Will the graph of a straight line always be a function? Give evidence. 4 Give algebraic evidence to show that the relation x2 + y2 = 9 is not a function.

B

FUNCTION NOTATION, DOMAIN AND RANGE

Function machines are sometimes used to illustrate how functions behave. x

For example:

So, if 4 is fed into the machine, 2(4) + 3 = 11 comes out.

I double the input and then add 3

2x + 3

The above ‘machine’ has been programmed to perform a particular function. If f is used to represent that particular function we can write: f is the function that will convert x into 2x + 3. So, f would convert 2 into 2(2) + 3 = 7 and ¡4 into 2(¡4) + 3 = ¡5. This function can be written as:

f : x `! 2x + 3 function f

such that

x is converted into 2x + 3

Two other equivalent forms we use are: f(x) = 2x + 3 or y = 2x + 3 f(x) is the value of y for a given value of x, i.e., y = f (x).

So,

Notice that for f (x) = 2x + 3, f(2) = 2(2) + 3 = 7 and f(¡4) = 2(¡4) + 3 = ¡5: Consequently,

f (2) = 7 indicates that the point (2, 7) lies on the graph of the function. f (¡4) = ¡5 indicates that the point (¡4, ¡5) also lies on the graph.

Likewise

y (2,¡7) ƒ(x)¡=¡2x¡+¡3 3 x 3

Note: ² f (x) is read as “f of x”.

(-4,-5)

² f is the function which converts x into f (x), i.e., f : x 7! f(x).

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Example 2 If f : x 7! 2x2 ¡ 3x, find the value of:

a f(5)

b f (¡4)

f (x) = 2x2 ¡ 3x = 2(5)2 ¡ 3(5) = 2 £ 25 ¡ 15 = 35

freplacing x by (5)g

a

f(5)

b

f(¡4) = 2(¡4)2 ¡ 3(¡4) = 2(16) + 12 = 44

freplacing x by (¡4)g

Example 3 If f (x) = 5 ¡ x ¡ x2 , find in simplest form: f(¡x) = 5 ¡ (¡x) ¡ (¡x)2 = 5 + x ¡ x2

a

b

a f(¡x)

b f(x + 2)

freplacing x by (¡x)g

f(x + 2) = 5 ¡ (x + 2) ¡ (x + 2)2 = 5 ¡ x ¡ 2 ¡ [x2 + 4x + 4] = 3 ¡ x ¡ x2 ¡ 4x ¡ 4 = ¡x2 ¡ 5x ¡ 1

freplacing x by (x + 2)g

EXERCISE 1B.1 1 If f : x 7! 3x + 2, find the value of: a f(0) b f (2) c f (¡1)

d

f(¡5)

e

f (¡ 13 )

2 If f : x 7! 3x ¡ x2 + 2, find the value of: a f(0) b f (3) c f (¡3)

d

f(¡7)

e

f ( 32 )

3 If f (x) = 7 ¡ 3x, find in simplest form: a f (a) b f(¡a) c f (a + 3) d f(b ¡ 1)

e f (x + 2)

f f(x + h)

4 If F (x) = 2x2 + 3x ¡ 1, find in simplest form: a F (x + 4) b F (2 ¡ x) c F (¡x) d F (x2 )

e F (x2 ¡ 1)

f F (x + h)

5 If G(x) =

2x + 3 : x¡4

evaluate i G(2) ii G(0) iii G(¡ 12 ) find a value of x where G(x) does not exist find G(x + 2) in simplest form find x if G(x) = ¡3:

a b c d

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6 f represents a function. What is the difference in meaning between f and f(x)?

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7 If the value of a photocopier t years after purchase is given by V (t) = 9650¡860t euros: a find V (4) and state what V (4) means b find t when V (t) = 5780 and explain what this represents c find the original purchase price of the photocopier. 8 On the same set of axes draw the graphs of three different functions f (x) such that f(2) = 1 and f (5) = 3: 9 Find a linear function f(x) = ax + b for which f (2) = 1 and f(¡3) = 11. 10 Given T (x) = ax2 + bx + c, find a, b and c if T (0) = ¡4, T (1) = ¡2 and T (2) = 6:

DOMAIN AND RANGE The domain of a relation is the set of permissible values that x may have. The range of a relation is the set of permissible values that y may have. For example: All values of x > ¡1 are permissible.

y

(1)

So, the domain is fx j x > ¡1g or x 2 [ ¡1, 1 [:

x

All values of y > ¡3 are permissible.

(-1,-3)

So, the range is fy j y > ¡3g or y 2 [ ¡3, 1 [. y

(2)

x can take any value.

(2,¡1)

So, the domain is fx j x is in R g or x 2 R .

x

y cannot be > 1. So, the range is fy j y 6 1g or y 2 ] ¡ 1, 1 ]. y

(3)

x can take all values except x = 2: So, the domain is fx j x 6= 2g.

y=1

Likewise, the range is fy j y 6= 1g. x x=2

The domain and range of a relation are often described using interval notation. For example:

The domain consists of all real x such that x > 3 and we write this as

y

fx j x > 3g or x 2 [ 3, 1 [:

range

2

(3, 2)

the set of all x

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Likewise the range would be fy j y > 2g or y 2 [ 2, 1 [.

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For the profit function alongside: ² the domain is fx j x > 0g or x 2 [ 0, 1 [

²

profit ($)

100 range

fy j y 6 100g or y 2 ] ¡ 1, 100 ]:

the range is

items made (x) 10

domain

Intervals have corresponding graphs. For example: fx j x > 3g or x 2 [ 3, 1 [

is read “the set of all x such that x is greater than or equal to 3” and has x 3 number line graph

fx j x < 2g or x 2 ] ¡1, 2 [

has number line graph

fx j ¡2 < x 6 1g or x 2 ] ¡2, 1 ]


fx j x 6 0 or x > 4g


2

-2

0

i.e., x 2 ] ¡1, 0 ] or ] 4, 1 [

Note:

a

1

x

4

x

for numbers between a and b we write a < x < b or x 2 ] a, b [ :

b

a

x

for numbers ‘outside’ a and b we write x < a or x > b i.e., x 2 ] ¡1, a [ È ] b, 1 [ where È means ‘or’.

b

Example 4 For each of the following graphs state the domain and range: y y a b (4,¡3) x

x (8,-2)

b

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fy j y > ¡1g or y 2 [¡1, 1[

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fx j x is in R g or x 2 R

Domain is Range is

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fy j y > ¡2g or y 2 [¡2, 1[

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Range is

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fx j x 6 8g or x 2 ] ¡ 1, 8]

Domain is

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(2,-1)

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EXERCISE 1B.2 1 Write down the domain and range for each of the following functions: a f(1, 3), (2, 5), (3, 7)g b f(¡1, 3), (0, 3), (2, 5)g c f(¡3, 1), (¡2, 1), (¡1, 1), (3, 1)g d f(x, y) j x2 + y2 = 4, x 2 Z , y > 0g 2 For each of the following graphs find the domain and range: y y a b c

y

(5,¡3)

(-1,¡1)

(0,¡2) x x

y = -1

x

x=2

d

e

y

y

f

y (-1,¡2)

x

(2,¡-2)

(1,¡-1)

f(x) =

x=2

x =-2

(-4,-3)

3 Find the domain and range of each of the following functions: c a f : x 7! 2x ¡ 1 b f : x 7! 3 p p f d f : x 7! x2 + 4 e y = x2 ¡ 4 g

x

-1

x

p 2¡x

3 f(x) = p 2x ¡ 5

h

i

f(x) = j3x ¡ 1j + 2 y=

5 x¡2

f : x 7! 2 +

3 5¡x

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3x ¡ 9 ¡x¡2

x2

f : x 7! x +

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y = x3 ¡ 3x2 ¡ 9x + 10

j

y = x2 + x¡2

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f : x 7! x4 + 4x3 ¡ 16x + 3

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f : x 7! 5x ¡ 3x2

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4 Use a graphics calculator to help sketch graphs of the following functions. Find the domain and range of each. p 1 a f(x) = x b f : x 7! 2 x p c f : x 7! 4 ¡ x d y = x2 ¡ 7x + 10

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INVESTIGATION 1

FLUID FILLING FUNCTIONS

When water is added at a constant rate to a cylindrical container, the depth of water in the container is a linear function of time.¡ This is because the volume of water added is directly proportional to the time taken to add it.¡ If water was not added at a constant rate the direct proportionality would not exist. water

The depth-time graph for the case of a cylinder would be as shown alongside:

depth

depth

DEMO

time

The question arises: ‘What changes in appearance of the graph occur for different shaped containers?’ Consider a vase of conical shape.

depth

time

What to do:

1 For each of the following containers, draw a ‘depth v time’ graph as water is added: a b c d e

2 Use the water filling demonstration to check your answers to question 1. 3 Write a brief report on the connection between the shape of a vessel and the corresponding shape of its depth-time graph. You may wish to discuss this in parts. For example, first examine cylindrical containers, then conical, then other shapes. Gradients of curves must be included in your report. 4 Draw possible containers as in question 1 which have the following ‘depth v time’ graphs: a b c d depth

depth

depth

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COMPOSITE FUNCTIONS, f¡±¡g

C

Given f : x 7! f (x) and g : x 7! g(x), then the composite function of f and g will convert x into f(g(x)). f ± g is used to represent the composite function of f and g. f ± g means “f following g” and (f ± g)(x) = f (g(x)) i.e., f ± g : x 7! f(g(x)). Consider f : x 7! x4

and g : x 7! 2x + 3.

f ± g means that g converts x to 2x + 3

and then

f converts (2x + 3) to (2x + 3)4 .

This is illustrated by the two function machines below. Notice how f is following g.

x g-function machine 2x + 3

I double and then add 3

f-function machine I raise the number to the power 4

2x + 3

4

Algebraically, if f(x) = x (f ± g)(x) = f (g(x)) = f (2x + 3) = (2x + 3)4

(2!\+\3)V

and g(x) = 2x + 3, then

fg operates on x firstg ff operates on g(x) nextg

and (g ± f )(x) = g(f (x)) = g(x4 ) = 2(x4 ) + 3 = 2x4 + 3

So, in general, f (g(x)) 6= g(f(x)). The ability to break down functions into composite functions is useful in differential calculus.

Example 5 Given f : x 7! 2x + 1 and g : x 7! 3 ¡ 4x find in simplest form: a (f ± g)(x) b (g ± f )(x) f(x) = 2x + 1 and g(x) = 3 ¡ 4x (f ± g)(x) = f (g(x)) = f (3 ¡ 4x) = 2(3 ¡ 4x) + 1 = 6 ¡ 8x + 1 = 7 ¡ 8x

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(g ± f)(x) = g(f(x)) = g(2x + 1) = 3 ¡ 4(2x + 1) = 3 ¡ 8x ¡ 4 = ¡8x ¡ 1

2(¢) + 1, 2(¤) + 1, 2(3x ¡ 4) + 1.

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f (¢) = f(¤) = f(3x ¡ 4) =

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Note: If f(x) = 2x + 1 then

5

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a

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If F (x) = (f ± g)(x), the domain of F is the domain of g excluding any values of x such that g(x) = u where f (u) is undefined.

Note:

EXERCISE 1C 1 Given f : x 7! 2x + 3 and g : x 7! 1 ¡ x, find in simplest form: a (f ± g)(x) b (g ± f)(x) c (f ± g)(¡3) 2 Given f : x 7! x2 and g : x 7! 2 ¡ x find (f ± g)(x) and (g ± f )(x). Find also the domain and range of f ± g and g ± f: 3 Given f : x 7! x2 + 1 and g : x 7! 3 ¡ x, find in simplest form: a (f ± g)(x) b (g ± f)(x) c x if (g ± f )(x) = f (x) 4 Functions f and g are defined as follows: f = f(0, 2), (1, 3), (2, 0), (3, 1)g b g±f Find a f ± g

g = f(0, 3), (1, 2), (2, 1), (3, 0)g c f ±f

5 f and g are defined as:

f = f(0, 3), (1, 0), (2, 1), (3, 2)g g = f(0, 1), (1, 2), (2, 3), (3, 0)g

Find f ± g

6 f and g are defined as:

f = f(0, 2), (1, 5), (2, 7), (3, 9)g g = f(2, 2), (5, 0), (7, 1), (9, 3)g

Find a f ± g b g±f

7 Given f(x) = a

x+3 x+2

x+1 , find in simplest form: x¡1 (g ± f )(x) c (g ± g)(x)

and g(x) =

(f ± g)(x)

b

In each case, find the domain of the composite function. 8

a If ax + b = cx + d for all values of x, show that a = c and b = d. Hint: If it is true for all x, it is true for x = 0 and x = 1. b Given f (x) = 2x + 3 and g(x) = ax + b and that (f ± g)(x) = x for all values of x, deduce that a = 12 and b = ¡ 32 . c Is the result in b true if (g ± f)(x) = x for all x?

D

SIGN DIAGRAMS

Sometimes we do not wish to draw a time-consuming graph of a function but wish to know when the function is positive, negative, zero or undefined. A sign diagram enables us to do this and is relatively easy to construct. A sign diagram consists of: ² ²

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a horizontal line which is really the x-axis positive (+) and negative (¡) signs indicating that the graph is above and below the x-axis respectively critical values, the numbers written below the line, which are the graph’s x-intercepts or where it is undefined.

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Consider the three functions given below. y = ¡2(x ¡ 1)2

y = (x + 2)(x ¡ 1)

Function

y

y=

y

y 1

Graph

4 x

x x

x -2

1

+ -2

Sign diagram

Notice that: ²

-

+

-

1

1

x

-

+ 0

x

x

A sign change occurs about a critical value for single factors such as (x + 2) and (x ¡ 1), indicating cutting of the x-axis.

²

No sign change occurs about the critical value for squared factors such as (x ¡ 1)2 , indicating touching of the x-axis.

²

shows a function is undefined at x = 0.

0

In general: ² ²

when a factor has an odd power there is a change of sign about that critical value when a factor has an even power there is no sign change about that critical value.

Example 6 a

Draw sign diagrams for:

b

y

-2

1

y

x=3

4

x

x

Qw_ x=-2

a

b -

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Example 7 a (x + 3)(x ¡ 1)

Draw a sign diagram of: a

(x + 3)(x ¡ 1) has critical values of ¡3 and 1.

b 2(2x + 5)(3 ¡ x) 2(2x + 5)(3 ¡ x) has critical values of ¡ 52 and 3.

b

+ -3

-

1

-\Tw_

x

We try any number > 1, e.g., x = 2 As (5)(1) > 0 we put a + sign here.

-

As the factors are ‘single’ the signs alternate giving: -

+

-3

x

We try any number > 3, e.g., x = 5 As 2(15)(¡2) < 0 we put a ¡ sign here.

As the factors are ‘single’ the signs alternate giving: +

3

+

x

-\Tw_

a 12 ¡ 3x2

b ¡4(x ¡ 3)2

1

3

x

Example 8 Draw a sign diagram of: a

12 ¡ 3x2 = ¡3(x2 ¡ 4) = ¡3(x + 2)(x ¡ 2) which has critical values of ¡2 and 2

b

¡4(x ¡ 3)2 has a critical value of 3.

-2

2

3

x

We try any number > 2 e.g., x = 3. As ¡3(5)(1) is < 0 we put a ¡ sign here. As the factors are ‘single’ the signs alternate. -

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We try any number > 3 e.g., x = 4. As ¡4(1)2 is < 0 we put a ¡ sign here. As the factor is ‘squared’ the signs do not change.

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Example 9 x¡1 : 2x + 1

Draw a sign diagram for x¡1 2x + 1

is zero when x = 1 and undefined when x = ¡ 12 : +

So, -\Qw_

1

x¡1 = 2x + 1

For x = 10,

x

9 21

>0

Since (x ¡ 1) and (2x + 1) are single factors the signs alternate. ) sign diagram is

+

-

-\Qw_

+ x

1

EXERCISE 1D 1 From the graphs below, draw corresponding sign diagrams: a b c y

y -1

x

2

y 3

x x

2

d

e

y

f

y -2

y

x x -2

x

2

1

g

h

i

y

y

x

y

j

-3

x -1

2

x=-1 y

k

x=1

l

y=2

x=-2 -1

4

y

x

x=2 1

x

x

x

2

y

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2 Draw a d g j m

sign diagrams for: (x + 4)(x ¡ 2) ¡(x + 1)(x ¡ 3) x2 ¡ 9 x2 ¡ 3x + 2 6 ¡ 16x ¡ 6x2

b e h k n

x(x ¡ 3) (2x ¡ 1)(3 ¡ x) 4 ¡ x2 2 ¡ 8x2 ¡2x2 + 9x + 5

c f i l o

3 Draw sign diagrams for: a (x + 2)2 d ¡(x ¡ 4)2 g 4x2 ¡ 4x + 1

b e h

(x ¡ 3)2 x2 ¡ 2x + 1 ¡x2 ¡ 6x ¡ 9

c f i

4 Draw sign diagrams for: x+2 a x¡1

x(x + 2) (5 ¡ x)(1 ¡ 2x) 5x ¡ x2 6x2 + x ¡ 2 ¡15x2 ¡ x + 2 ¡(x + 2)2 ¡x2 + 4x ¡ 4 ¡4x2 + 12x ¡ 9

b

x x+3

c

2x + 3 4¡x

d

4x ¡ 1 2¡x

e

3x x¡2

f

¡8x 3¡x

g

(x ¡ 1)2 x

h

4x (x + 1)2

i

(x + 2)(x ¡ 1) 3¡x

j

x(x ¡ 1) 2¡x

k

x2 ¡ 4 ¡x

l

3¡x 2x2 ¡ x ¡ 6

m

x2 ¡ 3 x+1

n

x2 + 1 x

o

x2 + 2x + 4 x+1

p

¡(x ¡ 3)2 (x2 + 2) x+3

q

¡x2 (x + 2) 5¡x

r

x2 + 4 (x ¡ 3)2 (x ¡ 1)

s

x¡5 +3 x+1

t

x¡2 ¡4 x+3

u

3x + 2 x ¡ 3 ¡ x¡2 x+3

E

INEQUALITIES (INEQUATIONS)

2x ¡ 1 x+3 6 are examples of inequalities. x 5 In this section we aim to find all values of the unknown for which the inequality is true.

2x + 3 > 11 ¡ x and

GROUP INVESTIGATION

SOLVING INEQUALITIES 3x + 2 > 4 was: 1¡x

Jon’s method of solving

If

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3x + 2 > 4, then 1¡x 3x + 2 > 4(1 ¡ x) 3x + 2 > 4 ¡ 4x ) 7x > 2 ) x > 27

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3x + 2 = 1¡x

However, Sarah pointed out that if x = 5, greater than 4. They concluded that there was something wrong with the method of solution.

17 ¡4

= ¡4 14

and ¡4 14 is not x=1

y y=4

A graph also highlighted an error.

x

It seems that the correct answer is 27 < x < 1.

Wu_

y=-3

1

y=

Questions:

3x + 2 1- x

1 At what step was Jon’s method wrong? 2 Suggest an algebraic method which does give the correct answer.

From the Investigation above you should have concluded that multiplying both sides of an inequality by an unknown can lead to incorrect results. We therefore need an alternative method. ² ² ² ²

To solve inequalities we use these steps:

² ² ² ²

Note:

if if if if

Make the RHS zero by shifting all terms to the LHS. Fully factorise the LHS. Draw a sign diagram for the LHS. Determine the values required from the sign diagram.

a > b and a > b and a > b and a > b > 0,

c 2 R , then a + c > b + c c > 0, then ac > bc c < 0, then ac < bc then a2 > b2

Example 10 a 3x2 + 5x > 2

Solve for x:

3x2 + 5x > 2 ) 3x + 5x ¡ 2 > 0 ) (3x ¡ 1)(x + 2) > 0

a

2

b x2 + 9 < 6x

fmaking RHS zerog ffully factorising LHSg

+

Sign diagram of LHS is

-2

x

Qe_

x 2 ] ¡ 1, ¡2 ] or x 2 [ 13 , 1 [ .

So, for LHS > 0,

x2 + 9 < 6x ) x ¡ 6x + 9 < 0 ) (x ¡ 3)2 < 0 Sign diagram of LHS is

b

+

2

fmake RHS zerog ffully factorising LHSg

+

+ 3

x

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So, the inequality is never true.

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Example 11 3x + 2 61 x¡4

a

Solve for x:

3x + 2 61 x¡4

a

b

3x + 2 ¡1 6 0 x¡4 µ ¶ µ ¶ 3x + 2 x¡4 ¡1 60 x¡4 x¡4 )

)

3x + 2 ¡ (x ¡ 4) 60 x¡4

)

+

-

)

+

-

0

x

qA_p_

1 , 1[. Thus, x 2 ] ¡ 1, 0 [ or [ 10

+

-3

1 6 10 x 1 ) ¡ 10 6 0 x ³x´ 1 ¡ 10 60 ) x x 1 ¡ 10x ) 60 x -

2x + 6 60 x¡4

)

1 6 10 x

b

x

4

x 2 [ ¡3, 4 [

Example 12 Solve for x: 3x + 1 3x > x¡1 x+1

) )

3x + 1 3x > x¡1 x+1 3x + 1 3x ¡ >0 x¡1 x+1

(3x + 1)(x + 1) ¡ 3x(x ¡ 1) >0 (x ¡ 1)(x + 1) 7x + 1 >0 (x ¡ 1)(x + 1)

) Sign diagram of LHS is:

-1

+

-\Qu_

1

+ x

Thus, x 2 ] ¡ 1, ¡ 17 [ or ]1, 1[ .

EXERCISE 1E

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(x ¡ 1)2 < 0 x2 > 3x 2x2 > 4 4x2 ¡ 4x + 1 < 0 2x2 ¡ 4x + 2 > 0 12x2 > 5x + 2

b e h k n q

25

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for x: (2 ¡ x)(x + 3) > 0 x2 > x x2 < 4 2x2 > x + 3 3x2 > 8(x + 2) 1 + 5x < 6x2

5

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1 Solve a d g j m p

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(2x + 1)(3 ¡ x) > 0 3x2 + 2x < 0 x2 + 4x + 4 > 0 6x2 + 7x < 3 6x2 + 1 6 5x 2x2 + 9 > 9x

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2 Solve for x: x+4 a >0 2x ¡ 1 2x d >1 x¡3 1 g > 100 x

e h

2 1 < 2x ¡ 5 x+7 x 1 > x+2 x

j m

n

F

f i

x2 + 5x 60 x2 ¡ 4 1 6x x

l o

2x ¡ 3 2x < x+2 x¡2

q

x+3 >0 2x + 3 x+2 1 < 2x ¡ 1 2 1¡x 0 x+3 4 x> x

k

x3 > x

p

x+1 ¡3 x¡1 x >5 2x ¡ 1

b

35

x2 61 3x ¡ 2

r

THE MODULUS FUNCTION The modulus of a real number x is its distance from 0 on the number line.

Because the modulus is a distance, it cannot be negative. 5

7 0

-5

7

So, the modulus of 7 is 7, which is written as j 7 j = 7 and the modulus of ¡5 is 5, which is written as j ¡5 j = 5: j x j is the distance of x from 0 on the number line.

Thus,

|x|

If x > 0

|x|

0

If x < 0

x

x

0

ALGEBRAIC DEFINITION ½ The modulus of x, j x j = y = j x j has graph

if x > 0 if x < 0

x ¡x

y y¡=¡|x| This branch is y = -x, x < 0.

This branch is y = x, x > 0. x

p p p p 72 = 49 = 7 and (¡5)2 = 25 = 5

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is an equivalent definition of j x j.

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Notice that

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Example 13 If a = ¡3 and b = 4 find: a j7 + aj b j7 + aj

a

j ab j j ab j

b

= j 7 + (¡3) j = j4j =4

¯ 2 ¯ ¯ a + 2b ¯

c

¯ 2 ¯ ¯ a + 2b ¯ ¯ ¯ = ¯ (¡3)2 + 2(4) ¯ = j9 + 8j = j 17 j = 17

c

= j (¡3)(4) j = j¡12 j = 12

Example 14 By replacing j x j with x for x > 0 and ¡x for x < 0, write the following functions without the modulus sign and hence graph each function: a f(x) = x ¡ j x j b f(x) = x j x j If x < 0, f (x) = x ¡ (¡x) = 2x. If x > 0, f (x) = x ¡ x = 0: ½ y = 2x for x < 0 So, we graph y = 0 for x > 0:

a

b If x > 0, f (x) = x(x) = x2 . If x < 0, f (x) = x(¡x) = ¡x2 : ½ y = x2 for x > 0 So, we graph y = ¡x2 for x < 0:

y

y

branch y = x 2

branch y = 0

x

x branch y = 2x

branch y = -x 2

EXERCISE 1F.1 1 If a = ¡2, b = 3, c = ¡4 find the value of: a jaj b jbj c e

ja ¡ bj

f

jaj ¡jbj

g

i

j a j2

j

a2

k

2 If x = ¡3, find the value of:

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j ab j

ja + bj ¯c¯ ¯ ¯ ¯ ¯ a

h

jaj + jbj

l

jcj jaj

95

100

Is j a ¡ b j = j a j ¡ j b j ?

50

b

75

c

¯ ¯ ¯ 2x + 1 ¯ ¯ ¯ ¯ 1¡x ¯

25

0

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75

j5j ¡ jxj

25

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b

75

25

Is j a + b j = j a j + j b j ?

0

a

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j5 ¡ xj

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a

jajjbj

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4 Copy and complete: j ab j

a

b

6

2

6

¡2

¡6

2

¡6

¡2

¯a¯ ¯ ¯ ¯ ¯ b

jajjbj

jaj jbj

What do you suspect? 5 Use the fact that j x j =

p x2

j ab j = j a j j b j

a

to prove that: ¯ a ¯ jaj ¯ ¯ b ¯ ¯= , b 6= 0 b jbj

c

ja ¡ bj = jb ¡ aj

6 Using j a j = a if a > 0 and ¡a if a < 0, write the following functions without modulus signs and hence graph each function: a y = jx ¡2j b y = jx+ 1j c y = ¡jxj d

y = jxj +x

e

g

y = jxj +jx ¡2j ¯ ¯ y = ¯ x2 ¡ 1 ¯

h

j

jxj x y = jxj ¡ jx ¡1j ¯ ¯ y = ¯ x2 ¡ 2x ¯ y=

k

f i l

y = x ¡ 2jxj ¯ ¯ y = ¯ x2 + 1 ¯ ¯ ¯ y = ¯ x2 + 3x + 2 ¯

MODULUS EQUATIONS From the previous exercise you should have discovered these properties of modulus: ²

j x j > 0 for all x 2

²

2

j x j = x for all x ¯ ¯ ¯ x ¯ jxj ¯ ¯= ¯ y ¯ j y j for all x and y, y 6= 0

²

²

j¡x j = j x j for all x

²

j xy j = j x j j y j for all x and y

²

j a ¡ b j = j b ¡ a j for all a and b.

It is clear that j x j = 2 has two solutions, x = 2 and x = ¡2. if j x j = a where a > 0, then x = §a.

In general,

We use this rule to solve modulus equations.

Example 15 a j 2x + 3 j = 7

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j 2x + 3 j = 7 2x + 3 = §7 ) 2x = 7 ¡ 3 or ¡7 ¡ 3 ) 2x = 4 or ¡10 ) x = 2 or ¡5

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100

a

b j 3 ¡ 2x j = ¡1

100

Solve for x:

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Example 16

¯ ¯ ¯ 3x + 2 ¯ ¯ ¯ ¯ 1¡x ¯=4

Solve for x:

¯ ¯ ¯ 3x + 2 ¯ ¯ ¯ ¯ 1¡x ¯=4 If

3x + 2 = §4 1¡x

)

3x + 2 = ¡4 1¡x then 3x + 2 = ¡4(1 ¡ x) ) 3x + 2 = ¡4 + 4x ) 6=x

3x + 2 =4 1¡x

If

then 3x + 2 = 4(1 ¡ x) ) 3x + 2 = 4 ¡ 4x ) 7x = 2 ) x = 27 So, x =

2 7

or 6:

j x j = j b j then x = §b.

Also notice that if

Example 17 j x + 1 j = j 2x ¡ 3 j

Solve for x:

If j x + 1 j = j 2x ¡ 3 j ,

x + 1 = §(2x ¡ 3) fusing property aboveg

then

If x + 1 = 2x ¡ 3 then 4 = x

If x + 1 = ¡(2x ¡ 3) then x + 1 = ¡2x + 3 ) 3x = 2 ) x = 23

So, x =

2 3

or 4:

EXERCISE 1F.2 1 Solve for x: a jxj = 3 d jx ¡ 1j = 3 g j 3x ¡ 2 j = 1 2 Solve for x: ¯ ¯ ¯ x ¯ ¯=3 a ¯¯ x¡1 ¯

b e h

j x j = ¡5 j3 ¡ xj = 4 j 3 ¡ 2x j = 3

c f i

jxj = 0 j x + 5 j = ¡1 j 2 ¡ 5x j = 12

b

¯ ¯ ¯ 2x ¡ 1 ¯ ¯ ¯ ¯ x+1 ¯=5

c

¯ ¯ ¯ x+3 ¯ 1 ¯ ¯ ¯ 1 ¡ 3x ¯ = 2

3 Solve for x: a

jx + 1j = j2 ¡ xj

b

jxj = j5 ¡ xj

c

j 3x ¡ 1 j = j x + 2 j

d

j 2x + 5 j = j 1 ¡ x j

e

j 1 ¡ 4x j = 2 j x ¡ 1 j

f

j 3x + 2 j = 2 j 2 ¡ x j

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4 Solve for x using i a graphical method ii an algebraic method: a j x + 2 j = 2x + 1 b j 2x + 3 j = 3 j x j ¡ 1 c j x ¡ 2 j = 25 x + 1

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MODULUS INEQUALITIES Notice that if j x j < 2 then x lies between ¡2 and 2 i.e., if j x j < 2, then ¡2 < x < 2: Likewise, if j x j > 2, then x > 2 or x < ¡2: ² ²

In general,

if j x j < k where k > 0, then ¡k < x < k if j x j > k where k > 0, then x < ¡k or x > k

Example 18 a j 3x ¡ 7 j < 10

Solve for x:

j 3x ¡ 7 j < 10 ) ¡10 < 3x ¡ 7 < 10 ) ¡3 < 3x < 17 ) ¡ 33 < x < 17 3

a

) x 2 ] ¡ 1, b

b j 3 ¡ 2x j > 4

17 3

fadding 7 to each partg

[

j 3 ¡ 2x j > 4 ) j 2x ¡ 3 j > 4 fusing j a ¡ b j = j b ¡ a jg ) 2x ¡ 3 > 4 or 2x ¡ 3 6 ¡4 ) 2x > 7 or 2x 6 ¡1 i.e., x 2 ] ¡ 1, ¡ 12 ] or [ 72 , 1 [ ) x > 72 or x 6 ¡ 12

Notice that a and b above could be solved by a different method. In j 3x ¡ 7 j < 10, 2 ) j 3x ¡ 7 j < 100 ) (3x ¡ 7)2 ¡ 102 < 0 [3x ¡ 7 + 10] [3x ¡ 7 ¡ 10] < 0 ) [3x + 3] [3x ¡ 17] < 0

)

+

The LHS has sign diagram:

Example 19

we see that both sides are non-negative fsquaring both sidesg fjaj2 = a2 for all ag

-

+

-1

Qd_U_

x

so x 2 ] ¡ 1,

17 3

[.

Find exactly where 2 j x ¡ 1 j > j 3 ¡ x j . 2 j x ¡ 1 j > j 3 ¡ x j are non-negative.

Both sides of the inequality

) we square both sides to get 4 j x ¡ 1 j2 > j 3 ¡ x j2 ) 4(x ¡ 1)2 ¡ (3 ¡ x)2 > 0 fas j a j2 = a2 for a 2 R g ) [2(x ¡ 1) + (3 ¡ x)] [2(x ¡ 1) ¡ (3 ¡ x)] > 0 ) (x + 1)(3x ¡ 5) > 0 The critical values are x = ¡1,

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x 2 ] ¡ 1, ¡1 ] or [ 53 , 1 [ (check with GCD)

)

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Sometimes a graphical solution is easier.

Example 20 Solve graphically: j 1 ¡ 2x j > x + 1: We draw graphs of y = j 1 ¡ 2x j and y = x + 1 on the same set of axes. ( 1 ¡ 2x for 1 ¡ 2x > 0, i.e., x 6 12 y = j 1 ¡ 2x j = ¡1 + 2x for 1 ¡ 2x < 0, i.e., x > 12 y

y = 1 - 2x

3

-1

Now j 1 ¡ 2x j > x + 1 when the graph of y = j 1 ¡ 2x j lies above y = x + 1, ) x < 0 or x > 2,

y = x +1

1

x

-1

2

Qw_

i.e., x 2 ] ¡1, 0 [ or x 2 ] 2, 1 [ .

EXERCISE 1F.3 1 Solve for x: a jxj < 4

b

jxj > 3

c

jx + 3j 6 1

d

jx + 4j > 2

e

j 2x ¡ 1 j < 3

f

j 3 ¡ 4x j > 2

g

j 2x + 1 j < 4

h

2 > jx ¡1j

i

j 3 ¡ 7x j < 4

j

j 2 ¡ 7x j > 5

k

j 5 ¡ 3x j 6 1

l

5 > j3 ¡xj

2 Solve: a jx ¡ 3j 6 4 e

b j 2x ¡ 1 j 6 3

jxj > j2 ¡ xj

3 Solve graphically: a j 2x ¡ 3 j < x

c j 3x + 1 j > 2 ¯ ¯ ¯ x ¯ ¯>3 ¯ g ¯ x¡2¯

3 j x j 6 j 1 ¡ 2x j

f

¯ ¯ c ¯ x2 ¡ x ¯ > 2

b 2x ¡ 3 < j x j

4 Graph the function f (x) =

d j 5 ¡ 2x j > 7 ¯ ¯ ¯ 2x + 3 ¯ ¯ ¯>2 h ¯ x¡1 ¯ d jxj ¡ 2 > j4 ¡ xj

jxj jxj , and hence find all values of x for which > ¡ 12 . x¡2 x¡2

a Draw the graph of y = j x + 5 j + j x + 2 j + j x j + j x ¡ 3 j. b P Q O

5

A

-5

-2

R 3

B

P, Q and R are factories which are 5, 2 and 3 km away from factory O respectively.

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A security service wishes to know where it should locate its premises along AB so

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that the total length of cable to the 4 factories is a minimum. i Explain why the total length of cable is given by j x + 5 j + j x + 2 j + j x j + j x ¡ 3 j where x is the position of the security service on AB. ii Where should the security service set up to minimise the length of cable to all 4 factories? What is the minimum length of cable? iii If a fifth factory at S, located 7 km right of O, also requires the security service, where should the security service locate its premises for minimum cable length? 6 Which of these is true? Give proof. a j x + y j 6 j x j + j y j for all x, y

G

b

j x ¡ y j > j x j ¡ j y j for all x, y

THE RECIPROCAL FUNCTION x 7!

1 x

or f (x) =

1 x

1 x

x

is defined as the reciprocal function.

It has graph:

Notice that: y

y=-x

y=

1 x

1 is undefined when x = 0 x 1 exists in the The graph of f (x) = x first and third quadrants only.

²

y=x

f (x) =

²

(1,¡1) x (-1,-1)

²

1 is symmetric about y = x x and y = ¡x

²

as as as as

²

f (x) =

f (x) =

! reads “approaches” or “tends to”

x ! 1, f (x) ! 0 (from above) x ! ¡1, f (x) ! 0 (from below) x ! 0 (from right), y ! 1 x ! 0 (from left), y ! ¡1

1 is asymptotic to the x-axis x and to the y-axis. The graph gets closer to the axes as it gets further from the origin.

EXERCISE 1G 1 2 4 , g(x) = , h(x) = on the same set of axes. x x x Comment on any similarities and differences.

1 Sketch the graphs of f(x) =

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1 2 4 2 Sketch the graphs of f(x) = ¡ , g(x) = ¡ , h(x) = ¡ on the same set of axes. x x x Comment on any similarities and differences.

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H

ASYMPTOTES OF OTHER RATIONAL FUNCTIONS 2x + 1 . x¡1 Notice that at x = 1, f(x) is undefined.

Consider the function f(x) = Its graph is:

As the graph approaches the vertical line x = 1, we say that x = 1 is a vertical asymptote.

y

y=2

Notice that:

f(1:001) = 3002 and f (0:999) = ¡2998

We write:

as x ! 1 (from the left), f (x) ! ¡1 as x ! 1 (from the right), f (x) ! 1

x -\Qw_ -1

or alternatively, as x ! 1¡ , f (x) ! ¡1 as x ! 1+ , f (x) ! 1 x=1

We also notice that f (1000) =

2001 ¡1999 + 2:003 and f (¡1000) = + 1:997 999 ¡1001

This indicates that y = 2 is a horizontal asymptote and we write: as x ! 1, y ! 2 (from above) or as x ! 1, y ! 2+ as x ! ¡1, y ! 2 (from below) as x ! ¡1, y ! 2¡ Notice that as jxj ! 1, f(x) ! 2 The sign diagram of y =

2x + 1 x¡1

+ -\Qw_

is

-

+ x

1

and can be used to discuss

the function near its vertical asymptote without having to graph the function.

Now consider the function f(x) = x + 1 + Its graph is:

x = 2 is a vertical asymptote. At x = 2, f (x) is undefined.

y

As x ! 2¡ , f(x) ! ¡1 as x ! 2+ , f(x) ! 1

So, as x ! 1, f(x) ! x + 1 (from above) as x ! ¡1, f(x) ! x + 1 (from below)

x

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Thus y = x + 1 is an oblique asymptote.

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y=x+1

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DISCUSSION

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ASYMPTOTES What would be the asymptotes of ² y=

2x2 (x ¡ 1)(x ¡ 4)

² y = x2 +

INVESTIGATION 2

1 ? x

FINDING ASYMPTOTES

Use the graphing package supplied or a graphics calculator to examine the following functions for asymptotes:

GRAPHING PACKAGE

b

y=

3x + 1 x+2

c

y=

3x ¡ 9 (x ¡ 2)(x + 1)

2x 2 x ¡4

e

y=

1¡x (x + 2)2

f

y=

x2 + 4 x¡1

x2 ¡ 1 x2 + 1

h

y=

x2 ¡ 6x + 5 (x + 1)2

i

y = x2 +

a

y = ¡1 +

d

y=

g

y=

3 x¡2

2 x

Further examples of asymptotic behaviour are seen in exponential, logarithmic and some trigonometric functions. A function may cross a horizontal or oblique asymptote, but never a vertical asymptote.

Note:

EXERCISE 1H

cyan

f (x) = 2 ¡

3 x+1

c

f : x 7!

x+3 (x + 1)(x ¡ 2)

d

f (x) = x +

2 x¡3

e

y=

f

y=

g

f : x 7! x + 2 +

h

g : x 7! 2x ¡

i

y=

j

y = x2 ¡

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4x ¡ 4x ¡ 5

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100

x2

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x2 ¡ 4 x2 + 4

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3 x¡2

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f : x 7!

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1 For the following functions: i determine the asymptotes ii discuss the behaviour of the function as it approaches its asymptotes iii sketch the graph of the function iv find the coordinates of all points where the function crosses its asymptotes.

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2x2 + 1 x2 ¡ 4 2x x2 + 1

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I

INVERSE FUNCTIONS

The operations of + and ¡, £ and ¥, squaring and finding the square root, are inverse operations as one undoes what the other does. p 82 = 8. For example, x + 3 ¡ 3 = x, x £ 3 ¥ 3 = x and A function y = f (x) may or may not have an inverse function. If y = f(x) has an inverse function, this new function f ¡1 (x) ² ² ²

must indeed be a function, i.e., satisfy the vertical line test must be the reflection of y = f (x) in the line y = x must satisfy the condition that f ¡1 : f (x) 7! x

The function y = x, defined as f : x 7! x, is the identity function. This means that, for any function f that has an inverse function f ¡1 , f ±f ¡1 and f ¡1 ±f must always equal the identity function. So, ( f ± f ¡1 )(x) = (f ¡1 ± f )(x) = x, so the inverse function undoes the effect of the function on x. If (x, y) lies on f , then (y, x) lies on f ¡1 . So reflecting the function in y = x has the algebraic effect of interchanging x and y. For example, f : y = 5x + 2 becomes f ¡1 : x = 5y + 2. Consider: y

y=ƒ(x)

y = f ¡1 (x) is the inverse of y = f(x) as ² it is also a function ² it is the reflection of y = f(x) in the oblique line y = x.

y=x y=ƒ -1(x)

1

x

1

y

This is the reflection of y = f(x) in y = x, but it is not the inverse function of y = f(x) as it fails the vertical line test.¡

y=ƒ(x) x

We say that the function y = f(x) does not have an inverse.

y=x

y = f -1 (x)

y=ƒ(x), x¡>¡0

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y=ƒ

However, consider the same function y = f (x) but with the domain x > 0. The function does now have an inverse function, as drawn alongside. y = f (x) subject to x 6 0 would also have an inverse function.

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Example 21 Consider f : x 7! 2x + 3. a On the same axes, graph f and its inverse function f ¡1 . b Find f ¡1 (x) using i coordinate geometry and the slope of f ¡1 (x) from a ii variable interchange. ¡1 c Check that (f ± f )(x) = (f ¡1 ± f)(x) = x a

b

f(x) = 2x + 3 passes through (0, 3) and (2, 7). ) f ¡1 (x) passes through (3, 0) and (7, 2). 1 2¡0 = . 7¡3 2 y¡0 1 So, its equation is = x¡3 2 x¡3 i.e., y = 2 x¡3 ) f ¡1 (x) = 2

i This line has slope

y

y=ƒ(x) (2,¡7)

y=x

y=ƒ -1(x)

(0,¡3)

(7,¡2) (3,¡0)

x

ii f is y = 2x + 3, so f ¡1 is x = 2y + 3 ) x ¡ 3 = 2y x¡3 x¡3 ) =y i.e., f ¡1 (x) = 2 2 (f ± f ¡1 )(x) = f (f ¡1 (x)) µ ¶ x¡3 =f 2 µ ¶ x¡3 =2 +3 2

c

(f ¡1 ± f )(x) = f ¡1 (f (x)) = f ¡1 (2x + 3) (2x + 3) ¡ 3 = 2 2x = 2 =x

and

=x

If ¦ includes point (a, b) then ¦-1 includes point (b, a).

EXERCISE 1I 1 For each of the following functions f i on the same axes graph y = x, f and f ¡1 ii find f ¡1 (x) using coordinate geometry and i iii find f ¡1 (x) using variable interchange:

a

f : x 7! 3x + 1

b

f : x 7!

x+2 . 4

2 For each of the following functions f i find f ¡1 (x) ii sketch y = f (x), y = f ¡1 (x) and y = x on the same axes iii show that f ¡1 ± f = f ± f ¡1 = x, the identity function:

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f : x 7! 2x + 5

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c

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3 Copy the graphs of the following functions and in each case include the graphs of y = x and y = f ¡1 (x) : a b c y y y 5 1

x

x

-3

4

-2

x

d

e

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f

y

y

2

1

(2,¡2)

x

x x

2

a Sketch the graph of f : x 7! x2 ¡ 4 and reflect it in the line y = x. b Does f have an inverse function? c Does f with restricted domain x > 0 have an inverse function?

4

5 Sketch the graph of f : x 7! x3

J

and its inverse function f ¡1 (x).

FUNCTIONS WHICH HAVE INVERSES

It is important to understand the distinction between one-to-one and many-to-one functions. A one-to-one function is any function where ² for each x there is only one value of y and ² for each y there is only one value of x. Functions that are one-to-one satisfy both the ‘vertical line test’ and the ‘horizontal line test’. ² ²

This means that:

no vertical line can meet the graph more than once no horizontal line can meet the graph more than once.

Functions that are not one-to-one are called many-to-one. While these functions must satisfy the ‘vertical line test’ they do not satisfy the ‘horizontal line test’, i.e., at least one y-value has more than one corresponding x-value.

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If the function y = f (x) is one-to-one, it will have an inverse function y = f ¡1 (x): If a function y = f(x) is many-to-one, it will not have an inverse function. Many-to-one functions can have inverse functions for a restricted part of the domain (see Example 22).

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Example 22 Consider f : x 7! x2 . a Explain why the function defined above does not have an inverse function. b Does f : x 7! x2 where x > 0 have an inverse function? c Find f ¡1 (x) for f : x 7! x2 , x > 0: d Sketch y = f (x), y = x and y = f ¡1 (x) for f in b and f ¡1 in c. f : x 7! x2 has domain x 2 R and is many-to-one. It does not pass the ‘horizontal line test’.

a

b

y

If we restrict the domain to x¡>¡0 or x¡2¡[0,¡1[, or in fact any domain which makes f one-to-one, it satisfies the ‘horizontal line test’ and so has an inverse function. y

f (x) = x 2 , x > 0

x x

f is defined by y = x2 , x > 0 ) f ¡1 is defined by x = y2 , y > 0 p ) y = § x, y > 0 p ) y= x p fas ¡ x is 6 0g p So, f ¡1 (x) = x

c

Note:

d

y

@=!X' !>0 @=~`! x

y=x

1 , x 6= 0, is called the reciprocal function. x It is said to be a self-inverse function as f = f ¡1 . 1 This is because the graph of y = is symmetrical about the line y = x. x Any function with a graph which is symmetrical about the line y = x must be a self-inverse function. The function f(x) =

EXERCISE 1J Note:

If the domain of a function is the set of all real numbers, then the statement x 2 R will be omitted.

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1 Which of the following functions have inverses? In each of these cases, write down the inverse function. a f(1, 2), (2, 4), (3, 5)g b f(¡1, 3), (0, 2), (1, 3)g c f(2, 1), (¡1, 0), (0, 2), (1, 3)g d f(¡1, ¡1), (0, 0), (1, 1)g

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a Show that f : x 7!

2

1 x

has an inverse function for all x 6= 0.

b Find f ¡1 algebraically and show that f is a self-inverse function. 3x ¡ 8 , x 6= 3 is a self-inverse function by: x¡3 reference to its graph b using algebra.

3 Show that f : x 7! a

4 The ‘horizontal line test’ says that: for a function to have an inverse function, no horizontal line can cut it more than once. a Explain why this is a valid test for the existence of an inverse function. b Which of the following functions have an inverse function? i ii iii y y y 1 x

x

-1

-2

2

1

x

(1,-1)

c For the functions in b which do not have an inverse, specify domains as wide as possible where each function does have an inverse. 5 Consider f : x 7! x2 where x 6 0. a Find f ¡1 (x). b Sketch y = f(x), y = x and y = f ¡1 (x) on the same set of axes. a Explain why f : x 7! x2 ¡ 4x + 3 is a function but does not have an inverse function. b Explain why f for x > 2 has an inverse function. p c Show that the inverse function of the function in b is f ¡1 (x) = 2 + 1 + x. d If the domain of f is restricted to x > 2, state the domain and range of i f ii f ¡1 . e Show that f ± f ¡1 = f ¡1 ± f = x, the identity function.

6

7 Given f : x 7! (x + 1)2 + 3 where x > ¡1 : a find the defining equation of f ¡1 b sketch, using technology, the graphs of y = f (x), y = x and y = f ¡1 (x) c state the domain and range of i f ii f ¡1 . 8 Consider the functions f : x 7! 2x + 5 and g : x 7! a

Find g ¡1 (¡1).

9 Given f : x 7! 5x a find i

(f ± g ¡1 )(x) = 9.

b

Solve for x if p and g : x 7! x :

f (2) ii

g ¡1 (4)

8¡x . 2

b solve the equation (g ¡1 ± f )(x) = 25.

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10 Given f : x 7! 2x and g : x 7! 4x ¡ 3 show that (f ¡1 ± g ¡1 )(x) = (g ± f )¡1 (x).

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11 Which of these functions is a self-inverse function, i.e., f ¡1 (x) = f (x)? 2 6 a f(x) = 2x b f (x) = x c f(x) = ¡x d f(x) = e f (x) = ¡ x x 12 Show that (f ± f ¡1 )(x) = (f ¡1 ± f )(x) = x for: x+3 a f(x) = 3x + 1 b f(x) = 4 13

a

y

A

b

x

c

y = f (x)

f(x) =

p x

B is the image of A under a reflection in the line y = x. If A is (x, f(x)), what are the coordinates of B under the reflection? Substitute your result from a into y = f ¡1 (x). What result do you obtain? Explain how to establish that f (f ¡1 (x)) = x also.

B

y = f -1( x)

c

REVIEW SET 1A 1 If f(x) = 2x ¡ x2

a f (2)

find:

b f(¡3)

c

f(¡ 12 )

2 For each of the following graphs determine: i the range and domain ii the x and y-intercepts iii whether it is a function iv if it has an inverse function y a b y 1

-1

5

x

-1

x

-3

-\Wl_T_

(2,-5)

3 Find a, b and c if f (0) = 5, f (¡2) = 21 and f (3) = ¡4 and f(x) = ax2 +bx+c. 4 Draw a sign diagram for: (3x + 2)(4 ¡ x)

a

b

x¡3 x2 + 4x + 4

5 If f(x) = 2x ¡ 3 and g(x) = x2 + 2, find: 6 Solve for x: x(x + 8) a 65 x+2

b

a

f (g(x))

b

g(f (x))

3 5 > x¡1 2x + 1

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7 Consider x 7! 2x ¡ 7. a On the same set of axes graph y = x, f and f ¡1 . b Find f ¡1 (x) using variable interchange. c Show that f ± f ¡1 = f ¡1 ± f = x, the identity function.

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8 Solve for x: ¯ ¯ ¯ 2x + 1 ¯ ¯ ¯=3 a ¯ x¡2 ¯

j 3x ¡ 2 j > j 2x + 3 j

b

4x + 1 x2 + x ¡ 6 a determine the asymptotes b discuss the behaviour of the function as it approaches its asymptotes c sketch the graph. x 10 Given f : x 7! 3x + 6 and h : x 7! , show that (f ¡1 ± h¡1 )(x) = (h ± f )¡1 (x). 3 9 For f : x 7!

REVIEW SET 1B 1 If g(x) = x2 ¡ 3x, find in simplest form

a

b

g(x + 1)

g(x2 ¡ 2)

2 For each of the following functions f (x) find f ¡1 (x) : 3 + 2x a f(x) = 7 ¡ 4x b f(x) = 5 3 For each of the following graphs, find the domain and range. y y a b y¡=¡(x-1)(x-5)

(1,-1) x x x¡=¡2

4 Copy the following graphs and draw the graph of each inverse function: y y a b 2 2

5

x

x

5 Draw a sign diagram for: b

x+9 +x x+5

b

x2 ¡ 3x ¡ 4 >0 x+2

6 Solve for x:

7 Find an f and a g function given that: p a f(g(x)) = 1 ¡ x2

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j 7 ¡ 3x j > 8

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8 Solve for x: a j 4x ¡ 2 j = j x + 7 j

µ

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2x2 + x 6 10

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a

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x2 ¡ 6x ¡ 16 x¡3

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3x ¡ 2 x2 ¡ 4 determine the asymptotes discuss the behaviour of the function as it approaches its asymptotes sketch the graph find the coordinates of all points where the function crosses its asymptotes.

9 For f (x) = 3 + a b c d

10 Given h : x 7! (x ¡ 4)2 + 3, x 2 [ 4, 1 [ a find the defining equation of h¡1

b

show that h ± h¡1 = h¡1 ± h = x

REVIEW SET 1C 1 If h(x) = 7 ¡ 3x: a find in simplest form h(2x ¡ 1)

find x if h(2x ¡ 1) = ¡2

b

2 For each of the following graphs find the domain and range: a b y

(-3,¡5)

y (2,¡5) y=x+2

x

x y=-3

(-1,-3) x=1

p 3 If f(x) = 1 ¡ 2x and g(x) = x: a find in simplest form i (f ± g)(x) ii (g ± f )(x) b What is the domain and range of f ± g and g ± f? 4 Solve for x: x2 ¡ 3 x¡1 x+2

j 2x ¡ 6 j > x + 3 x b Graph the function f(x) = and hence find all values of x for jxj + 1 x which > 13 jxj + 1 (x + 2)(x ¡ 3) a Draw a sign diagram for . x¡1 x2 + x ¡ 8 b Hence, solve for x: 1400 . p Using a graphics calculator with Y1 = 6 £ ( 2)^(n ¡ 1), we view a table of values: So, the first term to exceed 1400 is u17 where u17 = 1536. Note: Later we can solve problems like this one using logarithms.

EXERCISE 2D.1

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a Show that the sequence 12, ¡6, 3, ¡ 32 , ..... is geometric. b Find un and hence find the 13th term (as a fraction).

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a Show that the sequence 5, 10, 20, 40, ..... is geometric. b Find un and hence find the 15th term.

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1 For the geometric sequence with first two terms given, find b and c: a 2, 6, b, c, .... b 10, 5, b, c, ..... c 12, ¡6, b, c, .....

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SEQUENCES AND SERIES (Chapter 2)

4 Show that the sequence 8, ¡6, 4:5, ¡3:375, .... is geometric and hence find the 10th term as a decimal. p p 5 Show that the sequence 8, 4 2, 4, 2 2, .... is geometric. Hence find, in simplest form, the general term un . 6 Find k given that the following are consecutive terms of a geometric sequence: a 7, k, 28 b k, 3k, 20 ¡ k c k, k + 8, 9k 7 Find the general term un of the geometric sequence which has:

8

a

u4 = 24 and u7 = 192

b

u3 = 8 and u6 = ¡1

c

u7 = 24 and u15 = 384

d

u3 = 5 and u7 =

5 4

a Find the first term of the sequence 2, 6, 18, 54, .... which exceeds 10 000. p p b Find the first term of the sequence 4, 4 3, 12, 12 3, .... which exceeds 4800. c Find the first term of the sequence 12, 6, 3, 1:5, .... which is less than 0:0001 :

COMPOUND INTEREST Consider the following: You invest $1000 in the bank. You leave the money in the bank for 3 years. You are paid an interest rate of 10% p.a. The interest is added to your investment each year. An interest rate of 10% p.a. is paid, increasing the value of your investment yearly. Your percentage increase each year is 10%, so at the end of the year you will have 100% + 10% = 110% of the value at its start. This corresponds to a multiplier of 1:1 . After one year your investment is worth $1000 £ 1:1 = $1100 After two years it is worth $1100 £ 1:1 = $1000 £ 1:1 £ 1:1 = $1000 £ (1:1)2 = $1210

After three years it is worth $1210 £ 1:1 = $1000 £ (1:1)2 £ 1:1 = $1000 £ (1:1)3

This suggests that if the money is left in your account for n years it would amount to $1000 £ (1:1)n . Observe that:

u1 u2 u3 u4 .. .

= $1000 = u1 £ 1:1 = u1 £ (1:1)2 = u1 £ (1:1)3

= initial investment = amount after 1 year = amount after 2 years = amount after 3 years

un+1 = u1 £ (1:1)n = amount after n years un+1 = u1 £ r n

In general, we can use the compound interest formula

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n = number of years un+1 = amount after n years.

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u1 = initial investment r = growth multiplier

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Example 10 $5000 is invested for 4 years at 7% p.a. compound interest, compounded annually. What will it amount to at the end of this period? u5 = u1 £ r4

is the amount after 4 years

= 5000 £ (1:07)4

ffor a 7% increase 100% becomes 107%g

¼ 6553:98 f5000 So, it amounts to $6553:98 .

×

1:07 ^ 4 ENTER g

Example 11 How much should I invest now if I want the maturing value to be $10¡000 in 4¡years’ time, if I am able to invest at 8:5% p.a.¡ compounded annually?

u1 = ?, u5 = 10 000, r = 1:085 fusing un+1 = u1 £ rn g u5 = u1 £ r4 ) 10 000 = u1 £ (1:085)4 10 000 (1:085)4

)

u1 =

)

u1 ¼ 7215:74

f10 000

÷

1:085 ^ 4 ENTER g

So, you should invest $7215:74 now.

EXERCISE 2D.2 1

a What will an investment of $3000 at 10% p.a. compound interest amount to after 3 years? b What part of this is interest?

2 How much compound interest is earned by investing E20 000 at 12% p.a. if the investment is over a 4 year period? a What will an investment of 30 000 Yen at 10% p.a. compound interest amount to after 4 years? b What part of this is interest?

3

4 How much compound interest is earned by investing $80 000 at 9% p.a., if the investment is over a 3 year period? 5 What will an investment of 100 000 Yen amount to after 5 years if it earns 8% p.a. compounded twice annually? 6 What will an investment of $45 000 amount to after 21 months if it earns 7:5% p.a. compounded quarterly?

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7 How much money must be invested now if you require $20 000 for a holiday in 4 years’ time and the money can be invested at a fixed rate of 7:5% p.a. compounded annually?

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8 What initial investment is required to produce a maturing amount of $15 000 in 60 months’ time given that a fixed rate of 5:5% p.a. compounded annually is guaranteed? 9 How much should I invest now if I want a maturing amount of E25 000 in 3 years’ time and the money can be invested at a fixed rate of 8% p.a. compounded quarterly? 10 What initial investment is required to produce a maturing amount of 40 000 Yen in 8 years’ time if your money can be invested at 9% p.a., compounded monthly?

OTHER GEOMETRIC SEQUENCE PROBLEMS Example 12 The initial population of rabbits on a farm was 50. The population increased by 7% each week. a How many rabbits were present after: i 15 weeks ii 30 weeks? b How long would it take for the population to reach 500? We notice that u1 = 50 and r = 1:07 u2 = 50 £ 1:07 = the population after 1 week a

b

un+1 = u1 £ rn ) u16 = 50 £ (1:07)15 ¼ 137:95 i.e., 138 rabbits

i

ii and u31 = 50 £ (1:07)30 ¼ 380:61 i.e., 381 rabbits

un+1 = u1 £ (1:07)n after n weeks So, we need to find when 50 £ (1:07)n = 500. Trial and error on your calculator gives n ¼ 34 weeks or using the Equation Solver gives n ¼ 34:03

or by finding the point of intersection of Y1 = 50 £ 1:07^X and Y2 = 500 on a graphics calculator, the solution is ¼ 34:03 weeks.

EXERCISE 2D.3 1 A nest of ants initially consists of 500 ants. The population is increasing by 12% each week.

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2 The animal Eraticus is endangered. Since 1992 there has only been one colony remaining and in 1992 the population of the colony was 555. Since then the population has been steadily decreasing at 4:5% per year. Find: a the population in the year 2007 b the year in which we would expect the population to have declined to 50.

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SERIES A series is the addition of the terms of a sequence, i.e., u1 + u2 + u3 + :::: + un is a series. The sum of a series is the result when we perform the addition. Given a series which includes the first n terms of a sequence, its sum is Sn = u1 + u2 + u3 + :::: + un .

Example 13 For the sequence 1, 4, 9, 16, 25, .... a Write down an expression for Sn .

b Find Sn for n = 1, 2, 3, 4 and 5.

Sn = 12 + 22 + 32 + 42 + :::: + n2 fall terms are perfect squaresg

a

b S1 S2 S3 S4 S5

=1 = 1+4= 5 = 1 + 4 + 9 = 14 = 1 + 4 + 9 + 16 = 30 = 1 + 4 + 9 + 16 + 25 = 55

SIGMA NOTATION u1 + u2 + u3 + u4 + :::: + un can be written more compactly using sigma notation. P , which is called sigma, is the equivalent of capital S in the Greek alphabet. n P We write u1 + u2 + u3 + u4 + :::: + un as uk . So,

k=1

n P

uk reads “the sum of all numbers of the form uk where k = 1, 2, 3, ...., up to n”.

k=1

Example 14 7 P

a

Expand and evaluate:

b

(k + 1)

k=1 7 P

a

5 P 1 k 2 k=1 5 1 P k k=1 2

b

(k + 1)

k=1

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=2+3+4+5+6+7+8 = 35

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SEQUENCES AND SERIES (Chapter 2) n P

Note:

(ak + bk ) =

k=1

n P

ak +

k=1 n P

If c is a constant,

n P

bk

k=1 n P

cak = c

k=1

ak

n P

and

k=1

c = cn

k=1

EXERCISE 2E.1 1 For the following sequences: i write down an expression for Sn ii a 3, 11, 19, 27, .... b 42, 37, 32, 27, .... d

2, 3, 4 12 , 6 34 , ....

find S5 . c 12, 6, 3, 1 12 , ....

1, 12 , 14 , 18 , ....

e

f

1, 8, 27, 64, ....

2 Expand and evaluate: 4 P

a

(3k ¡ 5)

b

k=1

5 P

(11 ¡ 2k)

7 P

c

k=1

k(k + 1)

k=1

3 For un = 3n ¡ 1, write u1 +u2 +u3 +::::+ u20 the sum. 4 Show that: n P a c = cn

b

k=1

5

a

n P

cak = c

k=1 n P

Explain why

n P

k=1

b

n P

using sigma notation and evaluate

n P

k=1

(3k 2 + 4k ¡ 3) = 3

(ak + bk ) =

k=1 n P

10 £ 2k¡1

k=1

c

ak

5 P

d

k2 + 4

k=1

n P

n P

ak +

k=1

n P

bk

k=1

k ¡ 3n

k=1

n P n(n + 1) n(n + 1)(2n + 1) k2 = and 2 6 k=1 k=1 n P (k + 1)(k + 2). find in simplest form

Given that

k=

k=1

Check your answer in the case when n = 10.

ARITHMETIC SERIES An arithmetic series is the addition of successive terms of an arithmetic sequence. For example:

21, 23, 25, 27, ....., 49 is an arithmetic sequence.

So,

21 + 23 + 25 + 27 + ::::: + 49 is an arithmetic series.

SUM OF AN ARITHMETIC SERIES

If the first term is u1 and the common difference is d, then the terms are: u1 , u1 + d, u1 + 2d, u1 + 3d, etc.

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Suppose that un is the final term of an arithmetic series.

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So, Sn = u1 + (u1 + d) + (u1 + 2d) + :::: + (un ¡ 2d) + (un ¡ d) + un But Sn = un + (un ¡ d) + (un ¡ 2d) + :::: + (u1 + 2d) + (u1 + d) + u1

freversing themg

Adding these two equations vertically we get

2Sn = (u1 + un ) + (u1 + un ) + (u1 + un ) + :::: + (u1 + un ) + (u1 + un ) + (u1 + un ) | {z } n of these ) 2Sn = n(u1 + un ) n ) Sn = (u1 + un ) where un = u1 + (n ¡ 1)d 2 Sn =

So,

n (u1 + un ) 2

Sn =

or

n (2u1 + (n ¡ 1)d) 2

Example 15 Find the sum of 4 + 7 + 10 + 13 + :::: to 50 terms. The series is arithmetic with u1 = 4, d = 3 and n = 50: n So, S50 = 50 fusing Sn = (2u1 + (n ¡ 1)d)g 2 (2 £ 4 + 49 £ 3) 2 = 3875

Example 16 Find the sum of ¡6 + 1 + 8 + 15 + :::: + 141. The series is arithmetic with u1 = ¡6, d = 7 and un = 141: First we need to find n. Now un = u1 + (n ¡ 1)d = ) ¡6 + 7(n ¡ 1) = ) 7(n ¡ 1) = ) n¡1 = ) n=

141 141 147 21 22

Using Sn =

n 2 (u1 + un ), 22 2 (¡6 + 141)

S22 = = 11 £ 135 = 1485

EXERCISE 2E.2 1 Find the sum of: b 12 + 3 + 5 12 + 8 + :::: to 50 terms d 50 + 48 12 + 47 + 45 12 + :::: to 80 terms

a 3 + 7 + 11 + 15 + :::: to 20 terms c 100 + 93 + 86 + 79 + :::: to 40 terms 2 Find the sum of:

b 50 + 49 12 + 49 + 48 12 + :::: + (¡20)

a 5 + 8 + 11 + 14 + :::: + 101

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c 8 + 10 12 + 13 + 15 12 + :::: + 83

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3 Evaluate these arithmetic series: 10 P a (2k + 5) b

15 P

k=1

(k ¡ 50)

c

k=1

20 P k=1

µ

k+3 2

¶

4 An arithmetic series has seven terms. The first term is 5 and the last term is 53. Find the sum of the series. 5 An arithmetic series has eleven terms. The first term is 6 and the last term is ¡27. Find the sum of the series. 6

A bricklayer builds a triangular wall with layers of bricks as shown. If the bricklayer uses 171 bricks, how many layers did he build?

7 Each section of a soccer stadium has 44 rows with 22 seats in the first row, 23 in the second row, 24 in the third row, and so on. How many seats are there in: a row 44 b each section c the stadium which has 25 sections? 8 Find the sum of: a the first 50 multiples of 11 b the multiples of 7 between 0 and 1000 c the integers between 1 and 100 which are not divisible by 3. 1 2 n(n

9 Prove that the sum of the first n positive integers is

+ 1).

10 Consider the series of odd numbers 1 + 3 + 5 + 7 + :::: a What is the nth odd number un ? b Prove that “the sum of the first n odd numbers is n2 ”. c Check your answer to b by finding S1 , S2 , S3 and S4 . 11 Find the first two terms of an arithmetic sequence where the sixth term is 21 and the sum of the first seventeen terms is 0. 12 Three consecutive terms of an arithmetic sequence have a sum of 12 and a product of ¡80. Find the terms. Hint: Let the terms be x ¡ d, x and x + d. 13 Five consecutive terms of an arithmetic sequence have a sum of 40. The product of the middle and the two end terms is 224. Find the terms of the sequence.

GEOMETRIC SERIES A geometric series is the addition of successive terms of a geometric sequence. For example, So,

1, 2, 4, 8, 16, :::: , 1024 is a geometric sequence. 1 + 2 + 4 + 8 + 16 + :::: + 1024 is a geometric series.

SUM OF A GEOMETRIC SERIES

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If the first term is u1 and the common ratio is r, then the terms are: u1 , u1 r, u1 r2 , u1 r3 , .... etc.

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Sn = u1 + u1 r + u1 r2 + u1 r3 +

So,

u3

u2 and for r 6= 1,

Sn =

+ u1 rn¡2 + u1 rn¡1

::::

u4

u1 (rn ¡ 1) r¡1

un¡1 Sn =

or

un

u1 (1 ¡ rn ) . 1¡r

If Sn = u1 + u1 r + u1 r2 + u1 r3 + :::: + u1 rn¡2 + u1 rn¡1

Proof:

69

.... (1)

then rSn = (u1 r + u1 r2 + u1 r3 + u1 r4 + :::: + u1 rn¡1 ) + u1 rn ) rSn = (Sn ¡ u1 ) + u1 rn

ffrom (1)g

) rSn ¡ Sn = u1 r ¡ u1 n

) Sn (r ¡ 1) = u1 (rn ¡ 1) and so Sn =

u1 (rn ¡ 1) u1 (1 ¡ rn ) or for r 6= 1. r¡1 1¡r

Discuss the case r = 1.

Example 17 Find the sum of 2 + 6 + 18 + 54 + :::: to 12 terms. The series is geometric with u1 = 2, r = 3 and n = 12. 2(312 ¡ 1) 3¡1 = 531 440

fusing

So, S12 =

Sn =

u1 (rn ¡ 1) g r¡1

Example 18 Find a formula for Sn for 9 ¡ 3 + 1 ¡

1 3

+ :::: to n terms.

The series is geometric with u1 = 9, r = ¡ 13 , “n” = n. 9(1 ¡ u1 (1 ¡ r ) = 1¡r n

So, Sn = )

Sn =

27 4 (1

This answer cannot be simplified as we do not know if n is odd or even.

(¡ 13 )n ) 4 3

¡ (¡ 13 )n )

EXERCISE 2E.3 1 Find the sum of the following series: a 12 + 6 + 3 + 1:5 + :::: to 10 terms

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p1 2

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d 1¡

+ :::: to 15 terms

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c 6 ¡ 3 + 1 12 ¡

p p 7 + 7 + 7 7 + 49 + :::: to 12 terms

b

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+

1 2

¡

1 p 2 2

+ :::: to 20 terms

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2 Find a formula for Sn for: p p a 3 + 3 + 3 3 + 9 + :::: to n terms b 12 + 6 + 3 + 1 12 + :::: to n terms c 0:9 + 0:09 + 0:009 + 0:0009 + :::: to n terms d 20 ¡ 10 + 5 ¡ 2 12 + :::: to n terms 3 Evaluate these geometric series: 10 P 3 £ 2k¡1 b a

12 P

k=1

( 12 )k¡2

c

k=1

25 P

6 £ (¡2)k

k=1

4 Each year a salesperson is paid a bonus of $2000 which is banked into the same account. It earns a fixed rate of interest of 6% p.a. with interest being paid annually. The amount at the end of each year in the account is calculated as follows: A0 = 2000 A1 = A0 £ 1:06 + 2000 A2 = A1 £ 1:06 + 2000 etc. a Show that A2 = 2000 + 2000 £ 1:06 + 2000 £ (1:06)2 : b Show that A3 = 2000[1 + 1:06 + (1:06)2 + (1:06)3 ]: c Find the total bank balance after 10 years, assuming there are no fees or charges. 1 : 2n Find S1 , S2 , S3 , S4 and S5 in fractional form. From a guess the formula for Sn . u1 (1 ¡ rn ) Find Sn using Sn = . 1¡r Comment on Sn as n gets very large. What is the relationship between the given diagram and d?

5 Consider Sn = a b c d e

1 2

+

1 4

+

1 8

+

1 16

+ :::: +

Qr_

Qw_ qA_y_ Qi_ eA_w_

SUM OF AN INFINITE GEOMETRIC SERIES Sometimes it is necessary to consider Sn =

u1 (1 ¡ rn ) 1¡r

when n gets very large.

What happens to Sn in this situation? If j r j > 1, the series is said to be divergent and the sum becomes infinitely large. If ¡1 < r < 1, i.e., j r j < 1, then rn approaches 0 for very large n. This means that Sn will get closer and closer to

u1 . 1¡r S=

We say that the series converges and we write its sum as We call this the limiting sum of the series.

u1 1¡r

for j r j < 1

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This result can be used to find the value of recurring decimals.

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Example 19 0:¹7 = Write 0:¹7 as a rational number.

7 10

+

7 100

+

7 1000

+

7 10 000

+ ::::

which is a geometric series with infinitely many terms )

S=

So, 0:¹7 =

7 u1 = 10 1 = 1¡r 1 ¡ 10

7 9

7 9

EXERCISE 2E.4 1 Consider 0:¹3 =

3 10

3 100

+

+

3 1000

+ :::: which is an infinite geometric series.

a What are i u1 and b Using a, show that 0:¹3 = 13 . 2 Write as a rational number: 3 Use S =

u1 1¡r

ii

r?

a 0:4

b 0:16

c

0:312

to check your answers to Exercise 2E.3, questions 5d and 6c.

4 Find the sum of each of the following infinite geometric series: a 18 + 12 + 8 + :::::: b 18:9 ¡ 6:3 + 2:1 ¡ :::::: 5 Find each of the following: 1 3 P a k k=1 4

1 P

b

k=0

¡ ¢k 6 ¡ 25

6 Determine whether each of the following series is convergent. If so, find the sum of the series. If not, find the smallest value of n for which the sum of the first n terms of the series exceeds 100. a 18 ¡ 9 + 4:5 ¡ :::::: b 1:2 + 1:8 + 2:7 + :::::: 7 The sum of the first three terms of a convergent geometric series is 19. The sum of the series is 27. Find the first term and the common ratio. 8 The second term of a convergent geometric series is 85 . The sum of the series is 10. Show that there are two possible series and find the first term and the common ratio in each case. 9

A ball takes 1 second to hit the ground when dropped. It then takes 90% of this time to rebound to its new height and this continues until the ball comes to rest. a Show that the total time of motion is given by 1 + 2(0:9) + 2(0:9)2 + 2(0:9)3 + ::::: b Find Sn for the series in a. c How long does it take for the ball to come to rest?

ground

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This diagram is inaccurate as the motion is really up and down on the same spot. It has been separated out to help us visualise what is happening.

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F

MISCELLANEOUS PROBLEMS

EXERCISE 2F 1 Henk starts a new job selling TV sets. He hopes to sell 11 sets in the first week, 14 in the next, 17 in the next, and so on in arithmetic sequence. In what week does Henk hope to sell his 2000th TV set? 2 A computer is bought for $2795 and depreciates at a rate of 2% per month. After how many months will its value reduce to $500? 3 A geometric series has a second term of 6 and the sum of its first three terms is ¡14. Find its fourth term. 4 When a ball falls vertically off a table it rebounds 75% of its height after each bounce. If it travels a total distance of 490 cm, how high was the table top above the floor? 5 An arithmetic and a geometric sequence both have a first term of 1 and their second terms are equal. The 14th term of the arithmetic sequence is three times the third term of the geometric sequence. Find the twentieth term of each sequence. 6 Evaluate:

5 X

a

k=1

7 Find n given that: 1 µ X

8 Find x if

k=1

n X

a

3x 2

¶k¡1

12 X

b

k(k + 1)(k + 2)

100 £ (1:2)k¡3

k=6

b

(2k + 3) = 1517

k=1

n X

2 £ 3k¡1 = 177 146

k=1

= 4. n(3n + 11) . 2 b Find the twentieth term of the sequence.

9 The sum of the first n terms of an arithmetic sequence is a

Find its first two terms.

10 Mortgage repayments: $8000 is borrowed over a 2-year period at a rate of 12% p.a. Quarterly repayments are made and the interest is adjusted each quarter, which means that the amount repaid in the period is deducted and the interest is charged on the new amount owed. There are 2 £ 4 = 8 repayments and the interest per quarter is 12% 4 = 3%: At the end of the first quarter the amount owed, A1 , is given by $8000 £ 1:03 ¡ R, where R is the amount of each repayment. A2 = A1 £ 1:03 ¡ R At the end of the second quarter the amount owed, A2 , is given by: = ($8000 £ 1:03 ¡ R) £ 1:03 ¡ R = $8000 £ (1:03)2 ¡ 1:03R ¡ R

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a Find a similar expression for the amount owed at the end of the third quarter, A3 . b Write down an expression for the amount owed at the end of the 8th quarter, A8 , and hence deduce the value of R. Hint: What value do we want A8 to have? c If the amount borrowed at adjusted interest conditions is $P, the interest rate is r% per repayment interval, and there r m r ) £ 100 P (1 + 100 are m repayments, show that the : R= r m (1 + 100 ) ¡ 1 amount of each repayment is

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VON KOCH’S SNOWFLAKE CURVE

INVESTIGATION

C2

C1

C3

,

C4

,

,

, ......

In this investigation we consider a limit curve named after the Swedish mathematician Niels Fabian Helge von Koch (1870 - 1924). To draw Von Koch’s Snowflake curve we ² start with an equilateral triangle, C1 ² then divide each side into 3 equal parts ² then on each middle part draw an equilateral triangle ² then delete the side of the smaller triangle which lies on C1 . The resulting curve is C2 , and C3 , C4 , C5 , .... are found by ‘pushing out’ equilateral triangles on each edge of the previous curve as we did with C1 to get C2 . We get a sequence of special curves C1 , C2 , C3 , C4 , .... and Von Koch’s curve is the limiting case when n is infinitely large. Your task is to investigate the perimeter and area of Von Koch’s curve. What to do:

1 Suppose C1 has a perimeter of 3 units. Find the perimeter of C2 , C3 , C4 and C5 . becomes

Hint:

so 3 parts become 4 parts.

Remembering that Von Koch’s curve is Cn , where n is infinitely large, find the perimeter of Von Koch’s curve. 2 Suppose the area of C1 is 1 unit2 . Explain why the areas of C2 , C3 , C4 and C5 are A2 = 1 + A4 = 1 +

1 2 3 units 1 4 3 [1 + 9 +

A3 = 1 + 13 [1 + 49 ] units2 ( 49 )2 ] units2

A5 = 1 + 13 [1 +

4 9

+ ( 49 )2 + ( 49 )3 ] units2 .

Use your calculator to find An where n = 1, 2, 3, 4, 5, 6, 7, etc., giving answers which are as accurate as your calculator permits. What do you think will be the area within Von Koch’s snowflake curve? 3 Is there anything remarkable about your answers to 1 and 2? 4 Similarly, investigate the sequence of curves obtained by ‘pushing out’ squares on successive curves from the middle third of each side, i.e., the curves C1 , C2 , C3 , C4 , etc.

C3

C2

C1 ,

,

, ......

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Region contains 8 holes.

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REVIEW SET 2A 1 List the first four members of the sequences defined by: 3n + 2 a un = 3n¡2 b un = n+3

c

un = 2n ¡ (¡3)n

2 A sequence is defined by un = 68 ¡ 5n. a Prove that the sequence is arithmetic. b Find u1 and d. c Find the 37th term. d What is the first term of the sequence less than ¡200? a Show that the sequence 3, 12, 48, 192, .... is geometric. b Find un and hence find u9 .

3

4 Find k if 3k, k ¡ 2 and k + 7 are consecutive terms of an arithmetic sequence. 5 Find the general term of an arithmetic sequence given that u7 = 31 and u15 = ¡17. Hence, find the value of u34 . 6 A sequence is defined by un = 6( 12 )n¡1 . a Prove that the sequence is geometric. b Find u1 and r. c Find the 16th term to 3 significant figures. 7 Show that 28, 23, 18, 13, .... is arithmetic and hence find un and the sum Sn of the first n terms in simplest form. 8 Find k given that 4, k and k2 ¡ 1 are consecutive geometric terms. 9 Determine the general term of a geometric sequence given that its sixth term is 256 16 3 and its tenth term is 3 . 10 Find the sum of each of the following infinite geometric series: 4 8 a 1:21 ¡ 1:1 + 1 ¡ :::::: b 14 3 + 3 + 21 + :::::: 11 x + 3 and x ¡ 2 are the first two terms of a geometric series. Find the values of x for which the series converges. 12 The sum of the first two terms of a geometric series is 90. The third term is 24. a Show that there are two possible series and find the first term and the common ratio in each case. b Show that both series converge and find their respective sums. 13 a, b and c are consecutive terms of both an arithmetic and geometric sequence. What can be deduced about a, b and c?

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14 x, y and z are consecutive terms of a geometric sequence. If x + y + z = 73 and x2 + y 2 + z 2 = 91 9 , find the values of x, y and z.

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REVIEW SET 2B a Determine the number of terms in the sequence 24, 23 14 , 22 12 , ...., ¡36. b Find the value of u35 for the sequence in a. c Find the sum of the terms of the sequence in a.

1

2 Insert six numbers between 23 and 9 so that all eight numbers are in arithmetic sequence. 3 Find the formula for un , the general term of: a 86, 83, 80, 77, .... b 34 , 1, 76 , 97 , ....

c

100, 90, 81, 72:9, ....

Note: One of these sequences is neither arithmetic nor geometric. 4 Write down the expansion of: 5 Write in the form

n P

7 P

a

k2

8 k+3 P k=1 k + 2

b

k=1

(:::::) :

k=1

a

b

4 + 11 + 18 + 25 + :::: for n terms

6 Find the sum of: a 3 + 9 + 15 + 21 + :::: to 23 terms 7 Calculate:

b

2

k=1

+

1 8

+

1 16

+

1 32

+ :::: for n terms.

b 24 + 12 + 6 + 3 + :::: to 12 terms.

¶ 8 µ X 31 ¡ 3k

a

1 4

15 X

50(0:8)k¡1

k=1

8 Find the first term of the sequence 5, 10, 20, 40, .... which exceeds 10 000. 9 What will an investment of E6000 at 7% p.a. compound interest amount to after 5 years if the interest is compounded: a annually b quarterly c monthly? 10 Find the sum of each of the following infinite geometric series: p a 18 ¡ 12 + 8 ¡ :::::: b 8 + 4 2 + 4 + :::::: 11 2x and x ¡ 2 are the first two terms of a convergent geometric series. If the sum of the series is 18 7 , find x, clearly explaining why there is only one possible value. 12 Find

1 X ¡ ¢k¡1 5 25 . k=7

13 a, b, c, d and e are consecutive terms of an arithmetic sequence. Prove that a + e = b + d = 2c. 14 Find the sum of the n consecutive geometric terms which can be inserted between 1 and 2.

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15 Suppose an arithmetic sequence and a geometric sequence with common ratio r have r2 the same first two terms. Show that the third term of the geometric sequence is 2r ¡ 1 times the third term of the arithmetic sequence.

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REVIEW SET 2C 1 A geometric sequence has u6 = 24 and u11 = 768. Determine the general term of the sequence and hence find: b the sum of the first 15 terms. a u17 2 How many terms of the series 11 + 16 + 21 + 26 + :::: are needed to exceed a sum of 450? 3 Find the first term of the sequence 24, 8, 83 , 89 , .... which is less than 0:001 . a Determine the number of terms in the sequence 128, 64, 32, 16, ...., b Find the sum of these terms.

4

1 512 .

5 $12 500 is invested in an account which pays 8:25% p.a. compounded. Find the value of the investment after 5 years if the interest is compounded: a

b

half-yearly

monthly.

6 How much should be invested at a fixed rate of 9% p.a. compounded interest if you wish it to amount to $20 000 after 4 years with interest paid monthly? 7 In 2004 there were 3000 koalas on Koala Island. Since then, the population of koalas on the island has increased by 5% each year. a How many koalas were on the island in 2007? b In what year will the population first exceed 5000? 8 A ball bounces from a height of 2 metres and returns to 80% of its previous height on each bounce. Find the total distance travelled by the ball until it stops bouncing. 1 X

a Under what conditions will the series

9

b Find

1 X

50(2x¡1)k¡1 converge?

Explain!

k=1

50(2x ¡ 1)k¡1 if x = 0:3 :

k=1

3n2 + 5n . 2

10 The sum of the first n terms of a sequence is a Find the nth term. b Prove that the sequence is arithmetic.

11 a, b and c are consecutive terms of an arithmetic sequence. Prove that the following are also consecutive terms of an arithmetic sequence: 1 1 1 p p and p a b + c, c + a and a + b b p p , p c + a b+ c a+ b ¡

(111111::::::1) | {z }

(22222::::::2) | {z }

2n 1’s

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and 1111 ¡ 22 = 1089 = 332 :)

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(For example: 11 ¡ 2 = 9 = 32

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n 2’s

5

12 Show that

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Chapter

Exponentials

Contents:

A B C D E F G H

3

Index notation Evaluating powers Index laws Algebraic expansion and factorisation Exponential equations Graphs of exponential functions Growth and decay The natural exponential ‘e’

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Review set 3A Review set 3B Review set 3C

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78

EXPONENTIALS (Chapter 3)

We often deal with numbers that are repeatedly multiplied together. Mathematicians use indices or exponents to represent such expressions. For example, 5 £ 5 £ 5 = 53 . Indices have many applications in areas such as finance, engineering, physics, electronics, biology and computer science. Problems encountered in these areas may involve situations where quantities increase or decrease over time. Such problems are often examples of exponential growth or decay.

OPENING PROBLEM

LEGEND OF THE AMBALAPPUZHA PAAL PAYASAM

According to Hindu legend, Lord Krishna once appeared as a sage before the king who ruled a region of India, and challenged him to a game of chess. The prize if Lord Krishna won was based on the chessboard: that the king would provide him with a single grain of rice for the first square, two grains of rice for the second square, four grains of rice for the third square, and so on doubling the rice on each successive square on the board. Lord Krishna of course did win, and the king was most unhappy when he realised he owed more rice than there was in the world.

Consider the following questions: 1 Is there a function which describes the number of grains of rice on each square? 2 How many grains of rice would there be on the 40th square? 3 Using your knowledge of series, find the total number of grains of rice that the king owed.

A

INDEX NOTATION

Rather than writing 3 £ 3 £ 3 £ 3 £ 3, we can write such a product as 35 . 35 reads “three to the power of five” or “three with index five”. Thus 43 = 4 £ 4 £ 4 and 56 = 5 £ 5 £ 5 £ 5 £ 5 £ 5. If n is a positive integer, then an

is the product of n factors of a.

5

3 base

power, index or exponent

an = a £ a £ a £ a £ ...... £ a {z } | n factors

EXERCISE 3A 1 List the first six powers of: a 2 b 3 c 4 2 Copy and complete the values of these common powers:

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a 51 = :::: , 52 = :::: , 53 = :::: , 54 = :::: b 61 = :::: , 62 = :::: , 63 = :::: , 64 = :::: c 71 = :::: , 72 = :::: , 73 = :::: , 74 = ::::

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HISTORICAL NOTE Nicomachus discovered an interesting number pattern involving cubes and sums of odd numbers. Nicomachus was born in Roman Syria (now Jerash, Jordan) around 100 AD. He wrote in Greek and was a Pythagorean.

B

1 = 13 3 + 5 = 8 = 23 7 + 9 + 11 = 27 = 33 .. .

EVALUATING POWERS

So far we have only considered positive bases raised to a power. We will now briefly look at negative bases. Consider the statements below: (¡1)1 (¡1)2 (¡1)3 (¡1)4

(¡2)1 (¡2)2 (¡2)3 (¡2)4

= ¡1 = ¡1 £ ¡1 = 1 = ¡1 £ ¡1 £ ¡1 = ¡1 = ¡1 £ ¡1 £ ¡1 £ ¡1 = 1

= ¡2 = ¡2 £ ¡2 = 4 = ¡2 £ ¡2 £ ¡2 = ¡8 = ¡2 £ ¡2 £ ¡2 £ ¡2 = 16

From the patterns above we can see that: A negative base raised to an odd power is negative. A negative base raised to an even power is positive.

CALCULATOR USE Although different calculators vary in the appearance of keys, they all perform operations of raising to powers in a similar manner.

Power keys

x2

squares the number in the display.

^ 3

raises the number in the display to the power 3.

^ 5

raises the number in the display to the power 5. raises the number in the display to the power ¡4.

4

^ Example 1

a 65

Find, using your calculator:

cyan

Press: 6 ^

b

Press:

c

Press:

(

¡74

5 ENTER

Answer 7776

)

625

5

^ 4 ENTER

¡2401

^ 4 ENTER

7

c

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yellow

95

100

50

75

25

0

5

95

100

50

75

25

0

5

95

100

50

You will need to check if your calculator uses the same key sequence as in the examples.¡ If not, work out the sequence which gives you the correct answers.

75

25

0

5

95

100

50

75

25

0

5

Note:

a

b (¡5)4

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Example 2 a 5¡2

Find using your calculator, and comment on:

a

Press: 5 ^

b

Press: 1

b Answer 0:04

2 ENTER

÷

5

1 52

^ 2 ENTER

0:04 ¡2

The answers indicate that 5

1 = 2: 5

Notice the effect of the brackets in these examples.

EXERCISE 3B 1 Simplify, then use a calculator to check your answer: a

(¡1)5

b

(¡1)6

c

(¡1)14

d

(¡1)19

e

(¡1)8

f

¡18

g

¡(¡1)8

h

(¡2)5

i

¡25

j

¡(¡2)6

k

¡(¡5)4

l

¡(¡5)4

2 Use your calculator to find the value of the following, recording the entire display: a f

47 (¡8)6

b g

74 ¡86

c h

¡55 2:139

(¡5)5 ¡2:139

d i

3 Use your calculator to find the values of the following: 1 a 9¡1 b c 6¡2 91 1 e 3¡4 f g 170 34 What do you notice? 4 Consider 31 , 32 , 33 , 34 , 35 ...... .

e j

86 (¡2:13)9

d

1 62

h

(0:366)0

Look for a pattern and find the last digit of 3101 .

5 What is the last digit of 7217 ? 6 Answer the Opening Problem on page 78.

C

INDEX LAWS

(am )n = amn

²

a¡ n and an are reciprocals, i.e., a¡ n =

²

² a0 = 1 for all a 6= 0

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yellow

95

100

for all a 6= 0.

75

50

1 an

25

0

5

95

100

50

75

25

0

5

95

² (ab)n = an bn

100

50

25

0

5

95

75

²

cyan

£a =a n

µ ¶n a an ² = n (b 6= 0) b b

am = am ¡ n (a 6= 0) an

a

m

100

50

m +n

²

75

25

0

5

The following are laws of indices for m, n 2 Z :

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Example 3 Write as powers of 2: a 16

b

a

b

16 =2£2£2£2 = 24

1 16

1 16

c

1

d 4 £ 2n

c

1 = 20

d

1 24 = 2¡4 =

e

4 £ 2n = 22 £ 2n = 22+n

2m 8 2m 8

e =

2m 23

= 2m¡3

Example 4 a¡3 b2 c¡1

Write without negative indices: a¡3 b2 b2 c = c¡1 a3

1 , a3

fas a¡3 =

1 = c1 g c¡1

Example 5 1

Write

21¡n

1

in non-fractional form.

21¡n

= 2¡(1¡n) = 2¡1+n = 2n¡1

EXERCISE 3C.1 1 Write as powers of 2: a

4

b

g

2

h

1 4 1 2

e

32

f

j

1 8 1 64

k

128

l

1 32 1 128

27

d

1 27

e

3

f

1 3

1

j

243

k

1 243

c

8

d

i

64

c i

2 Write as powers of 3: a

9

b

g

81

h

1 9 1 81

3 Write as a single power of 2: a

2 £ 2a

b

4 £ 2b

c

f

2c 4

g

2m 2¡m

h

d

(2x+1 )2

e

21¡n

i

2x+1 2x

j

21¡x

27 £ 3d

e

9 £ 27t

j

9n+1 32n¡1

8 £ 2t 4

(21¡n )¡1 4x

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100

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25

i

0

9 27t

5

h

95

3 3y

100

g

50

3y 3

75

f

25

d

0

3 £ 9n

5

c

95

27a

100

b

50

9 £ 3p

75

a

25

0

5

95

100

50

75

25

0

5

4 Write as a single power of 3:

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5 Write without negative indices: a

ab¡2

b

f

a2 b¡1 c¡2

g

(ab)¡2 1 a¡3

c

(2ab¡1 )2

d

(3a¡2 b)2

e

a2 b¡1 c2

h

a¡2 b¡3

i

2a¡1 d2

j

12a m¡3

d

an b¡m

e

a¡n a2+n

6 Write in non-fractional form: 1 an

a

1

b

1

c

b¡n

32¡n

RATIONAL INDICES 1

1

1+1

Notice that a 2 £ a 2 = a 2 2 = a1 = a p p and a £ a = a also. 1

a2 =

So, 1

1

ffor index laws to be obeyedg

p a

fby direct comparisong

1

Likewise a 3 £ a 3 £ a 3 = a1 = a p p p and 3 a £ 3 a £ 3 a = a 1

suggests

a3 =

Thus in general,

an =

1

2

p 3 a p n a

2

where

p n a reads “the nth root of a”.

2

Notice also that a 3 £ a 3 £ a 3 = a2 ³ 2 ´3 ) a3 = a2 )

2

a3 = m

an =

In general,

fif (am )n = amn

is to be usedg

p 3 a2 : p n am

Example 6

p 3 2

a

p 3 2

a

Write as a single power of 2:

1 p 2

b

1 p 2

b

1

= 23 =

c

2£ 1 5

1

22

=2

2

= 25

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100

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p 5 4 1

¡1 2

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p 5 4

= (22 ) 5

1

=2

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Example 7

83

7

Use your calculator to evaluate 2 5 . Calculator:

7

For 2 5 press:

Answer:

÷

( 7

2 ^

5 )

¼ 2:639 015

ENTER

Example 8 4

¡2 3

Without using a calculator, write in simplest rational form: a 8 3 4

83

a

¡2 3

b

27

4

= (23 ) 3

= (33 )

3£ 4 3

¡2 3

3£¡ 2 3

f(am )n = amn g

=2 = 24 = 16

b 27

=3 = 3¡2 = 19

EXERCISE 3C.2 1 Write as a single power of 2: p 1 a 52 b p 5 2 p 4 f 2£ 32 g p 2 2 Write as a single power of 3: p 1 a 33 b p 3 3

c

p 2 2

d

h

p ( 2)3

i

c

p 4 3

d

3 Write the following in the form ax p p a 37 b 4 27 c 1 p 3 7

f

1 p 4 27

g

p 4 2 1 p 3 16 p 3 3

e j

e

1 p 3 2 1 p 8 1 p 9 3

where a is a prime number and x is rational: p p p 5 16 d 3 32 e 7 49 1 p 5 16

h

i

1 p 3 32

j

1 p 7 49

d

4

d

1 p 3 7

e

32 5

j

125

4 Use your calculator to find: 3

34

a

7

28

b

c

¡1 3

2

5 Use your calculator to evaluate in three different ways: p p p a 9 b 48 c 5 27

¡3 5

cyan

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yellow

h

8

95

100

50

¡4 3

75

0

¡3 2

16 4

25

9

3

c

5

g

95

0

5

95

100

50

75

¡1 2

100

4

83

50

f

5

b

75

42

25

3

a

25

0

5

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50

75

25

0

5

6 Without using a calculator, write in simplest rational form:

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3

d

25 2

i

27

¡4 3

2 ¡2 3

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D

ALGEBRAIC EXPANSION AND FACTORISATION (a + b)(c + d) = ac + ad + bc + bd (a + b)(a ¡ b) = a2 ¡ b2 (a + b)2 = a2 + 2ab + b2 (a ¡ b)2 = a2 ¡ 2ab + b2

EXPANSION LAWS

Example 9 ¡1 2

Expand and simplify: x ¡1 2

x

¡1 2

=x

3

1

¡1 2

(x 2 + 2x 2 ¡ 3x £x

3 2

¡1 2

+ x

3

1

¡1 2

(x 2 + 2x 2 ¡ 3x

)

) 1

¡1 2

£ 2x 2 ¡ x

¡1 2

1

feach term is £ by x¡ 2 g

£ 3x

= x1 + 2x0 ¡ 3x¡1 3 =x+2¡ x

fadding indicesg

Example 10

a

b (ex + e¡x )2

a (2x + 3)(2x + 1)

Expand and simplify:

(2x + 3)(2x + 1) = 2x £ 2x + 2x + 3 £ 2x + 3 = 22x + 4 £ 2x + 3 = 4x + 22+x + 3

b

(ex + e¡x )2 = (ex )2 + 2ex £ e¡x + (e¡x )2 = e2x + 2e0 + e¡2x = e2x + 2 + e¡2x

EXERCISE 3D.1 1 Expand and simplify:

cyan

magenta

) ¡1 2

x (x + 2x + 3x ¡1 2

(5x ¡ 2)(5x ¡ 4)

e

(3x ¡ 1)2

f

(4x + 7)2

h

(2x + 3)(2x ¡ 3)

i

(x 2 + x

k

(ex ¡ e¡x )2

l

(5 ¡ 2¡x )2

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black


1

)

1 2

c

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25

0

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95

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50

75

1

25

0

5

95

100

50

75

25

0

5

j

1

1 2

(3x + 2)(3x + 5)

100

(x 2 + 2)(x 2 ¡ 2) 2 (x + )2 x

3 2

b

50

g

1 2

x

75

(2x + 3)2

¡1 2

i

25

d

1

5¡x (52x + 5x )

0

2 Expand and simplify: a (2x + 1)(2x + 3)

f

1

x 2 (x 2 + x

h

5

2¡x (2x + 5)

c

3x (2 ¡ 3¡x )

e

95

ex (ex + 2)

100

g

2x (2x + 1)

b

50

d

x2 (x3 + 2x2 + 1)

75

a

(x2 + x + x )

¡1 2

1

¡1 2

)(x 2 ¡ x

)

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FACTORISATION AND SIMPLIFICATION Example 11 a 2n+3 + 2n

Factorise:

2n+3 + 2n = 2n 23 + 2n = 2n (23 + 1) = 2n £ 9

a

c

23n + 22n

c

23n + 22n = 22n 2n + 22n = 22n (2n + 1)

b 2n+3 + 8 2n+3 + 8 = 2n 23 + 8 = 8(2n ) + 8 = 8(2n + 1)

b

Example 12 a 4x ¡ 9

Factorise:

b 9x + 4(3x ) + 4

4x ¡ 9 = (2x )2 ¡ 32 = (2x + 3)(2x ¡ 3)

a

fdifference of two squaresg

9x + 4(3x ) + 4 = (3x )2 + 4(3x ) + 4 = (3x + 2)2

b

fcompare a2 + 4a + 4g fas a2 + 4a + 4 = (a + 2)2 g

Example 13 6n 3n

a

Simplify:

6n 3n

a =

4n 6n

b or

2n 3n 3n 1

=

6n 3n ¡ 6 ¢n

4n 6n =

2n 2n 2n 3n

=

4n 6n ¡ 4 ¢n

=

=

2n 3n

¡ 2 ¢n

b

3

= 2n

= 2n

or

6 3

Example 14

yellow

2m+2 ¡ 2m 2m 2m+2 ¡ 2m 2m m 2 2 2 ¡ 2m = 2m m 2 (4 ¡ 1) = 2m1 =3

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25

0

b

5

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0

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25

0

5

100

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25

3n + 6n 3n n 3 + 2n 3n = 3n n 3 (1 + 2n ) = 3n 1 = 1 + 2n

a

cyan

b

100

3n + 6n 3n

a

Simplify:

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Y:\HAESE\IB_HL-2ed\IB_HL-2ed_03\085IB_HL-2_03.CDR Thursday, 25 October 2007 9:47:11 AM PETERDELL

c c

2m+3 + 2m 9 2m+3 + 2m 9 2m 23 + 2m = 9 2m (8 + 1) = 91 m =2

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Example 15

4x + 2x ¡ 20 = 0

Solve for x:

4x + 2x ¡ 20 = 0 ) (2 ) + 2x ¡ 20 = 0 ) (2x ¡ 4)(2x + 5) = 0 ) 2x = 4 or 2x = ¡5 ) 2x = 22 ) x=2 x 2

fcompare a2 + a ¡ 20 = 0g fas a2 + a ¡ 20 = (a ¡ 4)(a + 5)g f2x cannot be negativeg

EXERCISE 3D.2 1 Factorise: a 52x + 5x d 5n+1 ¡ 5 g 3(2n ) + 2n+1

b e h

3n+2 + 3n 6n+2 ¡ 6 2n+2 + 2n+1 + 2n

c f i

en + e3n 4n+2 ¡ 16 3n+1 + 2(3n ) + 3n¡1

2 Factorise: a 9x ¡ 4 d 25 ¡ 4x g 9x + 10(3x ) + 25

b e h

4x ¡ 25 9x ¡ 4x 4x ¡ 14(2x ) + 49

c f i

16 ¡ 9x 4x + 6(2x ) + 9 25x ¡ 4(5x ) + 4

3 Factorise: a 4x + 9(2x ) + 18 d 9x + 4(3x ) ¡ 5

b e

4x ¡ 2x ¡ 20 25x + 5x ¡ 2

c f

9x + 9(3x ) + 14 49x ¡ 7x+1 + 12

4 Simplify: 12n a 6n 35x 7x

e

b

20a 2a

c

6b 2b

d

4n 20n

f

6a 8a

g

5n+1 5n

h

5n+1 5

5 Simplify: a

6m + 2m 2m

b

2n + 12n 2n

c

8n + 4n 2n

d

6n + 12n 1 + 2n

e

5n+1 ¡ 5n 4

f

5n+1 ¡ 5n 5n

g

2n ¡ 2n¡1 2n

h

2n + 2n¡1 2n + 2n+1

i

3n+1 ¡ 3n 3n + 3n¡1

µ

6 Simplify: 2 (n + 1) + 2 (n ¡ 1) n

cyan

magenta

3

n¡1 6

95

100

50

75

25

0

5

95

yellow

µ

¶

4x ¡ 2x ¡ 2 = 0 25x ¡ 23(5x ) ¡ 50 = 0

100

50

b e

25

0

5

95

100

50

75

25

0

5

95

100

50

75

25

0

5

7 Solve for x: a 4x ¡ 6(2x ) + 8 = 0 d 9x = 3x + 6

n

b

75

a

n

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¡3

n

c f

n+1 6

¶

9x ¡ 12(3x ) + 27 = 0 49x + 1 = 2(7x )

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E

EXPONENTIAL EQUATIONS

An exponential equation is an equation in which the unknown occurs as part of the index or exponent. For example: 2x = 8 and 30 £ 3x = 7 are both exponential equations. If 2x = 8, then 2x = 23 .

Thus x = 3, and this is the only solution.

If ax = ak , then x = k. So, if the base numbers are the same, we can equate indices.

Hence:

Example 16 x

x+2

a 2 = 16

Solve for x:

b 3

2x = 16 ) 2x = 24 ) x=4

a

=

Once we have the same base we then equate the indices.

1 27

3x+2 =

b

1 27

) 3x+2 = 3¡3 ) x + 2 = ¡3 ) x = ¡5

Example 17 4x = 8

a

Solve for x:

(22 )x = 23

)

a 4x = 8

1 3 ¡1

)

(32 )x¡2 = 3

) )

32(x¡2) = 3¡1 2x ¡ 4 = ¡1 ) 2x = 3 ) x = 32

) 22x = 23 ) 2x = 3 ) x = 32

1 3

b 9x¡2 =

9x¡2 =

b

Remember to use the index laws correctly!

EXERCISE 3E 1 Solve for x: a 2x = 2 1 2

e

2x =

i

2x¡2 =

1 4

b

2x = 4

c

3x = 27

f

3x =

1 3

g

2x =

j

3x+1 =

k

2x+1 = 64

1 27

1 8

d

2x = 1

h

2x+1 = 8

l

21¡2x =

1 2

cyan

9x¡3 = 3 ( 14 )1¡x = 8

yellow

95

n

50

j

1 3

1 7

d

49x =

8x+2 = 32

h

81¡x =

k

( 12 )x+1 = 2

l

( 13 )x+2 = 9

o

( 17 )x = 49

p

( 12 )x+1 = 32

100

g

1 5

75

25x =

25

50

25

0

5

100

magenta

9x =

0

4x = 8¡x

c

5

m

1 2

f

95

42x¡1 =

8x =

100

i

1 8

1 4

b

75

4x =

95

e

50

4x = 32

75

a

25

0

5

95

100

50

75

25

0

5

2 Solve for x:

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1 4

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88


3 Solve for x: 42x+1 = 81¡x

a

b

92¡x = ( 13 )2x+1

c

2x £ 81¡x =

b

7 £ 2x = 56

c

3 £ 2x+1 = 24

e

4 £ ( 13 )x = 36

f

5 £ ( 12 )x = 20

4 Solve for x: a 3 £ 2x = 24 12 £ 3¡x =

d

4 3

1 4

F GRAPHS OF EXPONENTIAL FUNCTIONS We have learned to deal with bn where n 2 Q , i.e., n is any rational number. But what about bn where n 2 R , i.e., where n is not necessarily a rational? To answer this question, we can look at graphs of exponential functions. y

The most simple general exponential function has the form y = bx where b > 0, b 6= 1.

8

For example, y = 2x is an exponential function.

6

Table of values:

y = 2x

4

x

¡3

¡2

¡1

0

1

2

3

y

1 8

1 4

1 2

1

2

4

8

2 1 ¡10

We notice that for x = ¡10, say, y = 2

¼ 0:001

-3 -2 -1

x 1

2

3

Also when x = ¡50, y = 2¡50 ¼ 8:88 £ 10¡16 So, it appears that as x becomes large and negative, the graph of y = 2x approaches the x-axis from above it. We say that y = 2x is ‘asymptotic to the x-axis’, or ‘y = 0 is a horizontal asymptote’.

INVESTIGATION 1

EXPONENTIAL GRAPHS

We will investigate families of exponential functions.

GRAPHING PACKAGE

TI C

What to do:

a On the same set of axes, use a graphing package or graphics calculator to graph the following functions: y = 2x , y = 3x , y = 10x , y = (1:3)x .

cyan

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yellow

95

100

50

75

25

0

5

95

100

50

75

25

0

5

95

100

functions in a are all members of the family y = bx : What effect does changing b values have on the shape of the graph? What is the y-intercept of each graph? What is the horizontal asymptote of each graph?

50

25

0

5

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5

b The i ii iii

75

1

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89

a On the same set of axes, use a graphing package or graphics calculator to graph the following functions: y = 2x , y = 2x + 1, y = 2x ¡ 2. b The functions in a are all members of the family y = 2x + d where d is a constant. i What effect does changing d values have on the position of the graph? ii What effect does changing d values have on the shape of the graph? iii What is the horizontal asymptote of each graph? iv What is the horizontal asymptote of y = 2x + d? c To graph y = 2x + d from y = 2x what transformation is used?

3

a On the same set of axes, use a graphing package or graphics calculator to graph the following functions: y = 2x , y = 2x¡1 , y = 2x+2 , y = 2x¡3 . b The i ii iii

functions in a are all members of the family y = 2x¡c : What effect does changing c values have on the position of the graph? What effect does changing c values have on the shape of the graph? What is the horizontal asymptote of each graph?

c To graph y = 2x¡c 4

from y = 2x , what transformation is used?

a On the same set of axes, use a graphing package or graphics calculator to graph the functions y = 2x and y = 2¡x . b

i What is the y-intercept of each graph? ii What is the horizontal asymptote of each graph? iii What transformation moves y = 2x to y = 2¡x ?

a On the same set of axes, use a graphing package or graphics calculator to graph the following functions: i y = 2x , y = 3 £ 2x , y = 12 £ 2x ii y = ¡2x , y = ¡3 £ 2x , y = ¡ 12 £ 2x

5

b The functions in a are all members of the family y = a £ 2x where a is a constant. Comment on the effect on the graph when i a > 0 ii a < 0. c What is the horizontal asymptote of each graph? Why? From your investigation you should have discovered that: For the general exponential function y = a £ bx¡c + d I I I I

b controls how steeply the graph increases or decreases c controls horizontal translation d controls vertical translation and y = d is the equation of the horizontal asymptote. ² If a > 0, b > 1 ² If a > 0, 0 < b < 1 the function is the function is increasing. decreasing.

cyan

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95

If a < 0, 0 < b < 1 the function is increasing.

100

50

75

25

0

5

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100

50

75

25

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5

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100

50

²

If a < 0, b > 1 the function is decreasing.

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75

25

0

5

²

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We can sketch reasonably accurate graphs of exponential functions using:

² ² ²

All exponential graphs are similar in shape and have a horizontal asymptote.

the horizontal asymptote the y-intercept two other points, say when x = 2, x = ¡2

Example 18 Sketch the graph of y = 2¡x ¡ 3: For y = 2¡x ¡ 3 the horizontal asymptote is y = ¡3

y

0

When x = 0, y = 2 ¡ 3 =1¡3 = ¡2

x

) the y-intercept is ¡2

2

When x = 2, y = 2¡2 ¡ 3 = 14 ¡ 3

-2

y¡=¡2 -x-3

H.A. y=-3

= ¡2 34 When x = ¡2, y = 22 ¡ 3 = 1

We now have a well-defined meaning for bn where b, n 2 R because simple exponential functions have smooth increasing or decreasing graphs.

EXERCISE 3F

y x

1 Given the graph of y = 2 we can find approximate values of 2x for various x values. For example: (see point A) I 21:8 ¼ 3:5 2:3 (see point B) I 2 ¼5

y = 2x

5

B

4 A 3 2

Use the graph to determine approximate values of: p 1 b 20:8 a 2 2 or 2 c 21:5 d 2¡1:6 p 2

e

p

2¡

f

2

1

-2

2

-1

1

H.A. y=0 x 2

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2 Draw freehand sketches of the following pairs of graphs using your observations from

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the previous investigation: a y = 2x and y = 2x ¡ 2 c y = 2x and y = 2x¡2

y = 2x y = 2x

b d

91

and y = 2¡x and y = 2(2x ) GRAPHING PACKAGE

3 Check your answers to 2 using technology. 4 Draw freehand sketches of the following pairs of graphs: a y = 3x and y = 3¡x b y = 3x and y = 3x + 1 d y = 3x and y = 3x¡1 c y = 3x and y = ¡3x 5 Sketch the graphs of: a y = 2x + 1

y = 3 ¡ 2¡x p 6 Use your GDC to graph the functions in question 5 above and find y when x = 2: y = 2 ¡ 2x

b

c

y = 2¡x + 3

d

7 For the graphs of the functions in question 5 above, discuss the behaviour of y as x ! §1. Hence determine the horizontal asymptotes for each graph.

G

GROWTH AND DECAY

In this exercise we will examine situations where quantities are either increasing or decreasing exponentially. These situations are known as growth and decay, and occur frequently in the world around us. For example, populations of animals, people, and bacteria usually grow in an exponential way. Radioactive substances and items that depreciate usually decay exponentially.

GROWTH Consider a population of 100 mice which under favourable conditions is increasing by 20% each week. To increase a quantity by 20%, we multiply it by 120% or 1:2.

400

So, if Pn is the population after n weeks, then

200

P0 P1 P2 P3

= 100 fthe original populationg = P0 £ 1:2 = 100 £ 1:2 = P1 £ 1:2 = 100 £ (1:2)2 = P2 £ 1:2 = 100 £ (1:2)3 , etc

Pn

300

100 n (weeks)

1

2

3

4

5

6

and from this pattern we see that Pn = 100 £ (1:2)n . Alternatively:

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This is an example of a geometric sequence and we could find the rule to generate it. Clearly r = 1:2 and so as Pn = P0 rn , then Pn = 100 £ (1:2)n for n = 0, 1, 2, 3, ....

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Example 19 An entomologist monitoring a grasshopper plague notices that the area affected by the grasshoppers is given by An = 1000 £ 20:2n hectares, where n is the number of weeks after the initial observation. a Find the original affected area. b Find the affected area after i 5 weeks ii 10 weeks. c Find the affected area after 12 weeks. d Draw the graph of An against n. a

A0 = 1000 £ 20 = 1000 £ 1 = 1000 ) original area was 1000 ha.

b

i

c

A12 = 1000 £ 20:2£12 = 1000 £ 22:4 fPress: 1000 ¼ 5278 d ) after 12 weeks, area affected is about 5300 ha.

A5 = 1000 £ 21 = 2000 i.e., area is 2000 ha.

ii A10 = 1000 £ 22 = 4000 i.e., area is 4000 ha.

×

2 ^ 2:4 ENTER g

A (ha)

c

6000

b ii 4000

bi 2000

a

n (weeks) 2

4

6

8 10 12 14

EXERCISE 3G.1 1 The weight Wt of bacteria in a culture t hours after establishment is given by Wt = 100 £ 20:1t grams. Find: a the initial weight b the weight after i 4 hours ii 10 hours iii 24 hours. c Sketch the graph of Wt against t using the results of a and b only. d Use technology to graph Y1 = 100 £ 20:1X and check your answers to a, b and c. 2 A breeding program to ensure the survival of pygmy possums was established with an initial population of 50 (25 pairs). From a previous program, the expected population Pn in n years’ time is given by Pn = P0 £ 20:3t .

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What is the value of P0 ? What is the expected population after: i 2 years ii 5 years iii 10 years? Sketch the graph of Pn against n using a and b only. Use technology to graph Y1 = 50 £ 20:3X and check your answers in b.

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3 The speed Vt of a chemical reaction is given by Vt = V0 £ 20:05t where t is the temperature in o C. Find: b the speed at 20o C a the speed at 0o C c the percentage increase in speed at 20o C compared with the speed at 0o C. µ ¶ V50 ¡ V20 d Find £ 100%. What does this calculation represent? V20 4 A species of bear is introduced to a large island off Alaska where previously there were no bears. 6 pairs of bears were introduced in 1998. It is expected that the population will increase according to Bt = B0 £ 20:18t where t is the time since the introduction. b Find the expected bear population in 2018. a Find B0 . c Find the expected percentage increase from 2008 to 2018.

DECAY Now consider a radioactive substance of original weight 20 grams which decays or reduces by 5% each year. The multiplier is now 95% or 0:95. So, if Wn is the weight after n years, then: W0 W1 W2 W3 .. .

25 W (grams) n 20

= 20 grams = W0 £ 0:95 = 20 £ 0:95 grams = W1 £ 0:95 = 20 £ (0:95)2 grams = W2 £ 0:95 = 20 £ (0:95)3 grams

15

10 5

W20 = 20 £ (0:95)20 ¼ 7:2 grams .. . 100

W100 = 20 £ (0:95)

10

¼ 0:1 grams

n (years) 20

and from this pattern we see that Wn = 20 £ (0:95)n . Alternatively: Once again we have a geometric sequence with W0 = 20 and r = 0:95, and consequently Wn = 20 £ (0:95)n for n = 0, 1, 2, 3, ....

Example 20

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When a diesel-electric generator is switched off, the current dies away according to the formula I(t) = 24 £ (0:25)t amps, where t is the time in seconds. a Find I(t) when t = 0, 1, 2 and 3. b What current flowed in the CD player at the instant when it was switched off? c Plot the graph of I(t) for t > 0 using the information above. d Use your graph and/or technology to find how long it takes for the current to reach 4 amps.

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a

I(t) = 24 £ (0:25)t amps I(1) = 24 £ (0:25)1 = 6 amps

I(0) = 24 £ (0:25)0 = 24 amps b

25 20 15 10 5 0

d

I(3) = 24 £ (0:25)3 = 0:375 amps

) 24 amps of current flowed.

When t = 0, I(0) = 24

c

I(2) = 24 £ (0:25)2 = 1:5 amps

I (amps)

1

3

2

t (seconds) 4

From the graph above, the approximate time to reach 4 amps is 1:3 seconds. or By finding the point of intersection of Y1 = 24 £ (0:25)^ X and Y2 = 4 on a graphics calculator, the solution is ¼ 1:29 seconds.

Example 21 The weight of radioactive material remaining after t years is given by Wt = W0 £ 2¡0:001t grams. a Find the original weight. b Find the percentage remaining after 200 years. a

When t = 0, W0 = W0 £ 20 = W0 ) W0 is the original weight.

b

When t = 200, W200 = W0 £ 2¡0:001£200 = W0 £ 2¡0:2 ¼ W0 £ 0:8706 ¼ 87:06% of W0

) 87:1% remains.

EXERCISE 3G.2

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1 The weight of a radioactive substance t years after being set aside is given by W (t) = 250 £ (0:998)t grams. a How much radioactive substance was put aside? b Determine the weight of the substance after: i 400 years ii 800 years iii 1200 years. c Sketch the graph of W (t) for t > 0, using the above information. d Use your graph or graphics calculator to find how long it takes for the substance to decay to 125 grams.

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2 The temperature T of a liquid which has been placed in a refrigerator is given by T (t) = 100 £ 2¡0:02t o C where t is the time in minutes. Find: a the initial temperature b the temperature after: i 15 minutes ii 20 minutes iii 78 minutes. c Sketch the graph of T (t) for t > 0 using a and b only. 3 The weight Wt grams of radioactive substance remaining after t years is given by Wt = 1000 £ 2¡0:03t grams. Find: a the initial weight b the weight after: i 10 years ii 100 years iii 1000 years. c Graph Wt against t using a and b only. 4 The weight Wt of radioactive uranium remaining after t years is given by the formula Wt = W0 £ 2¡0:0002t grams, t > 0. Find: a the original weight

H

b

the percentage weight loss after 1000 years.

THE NATURAL EXPONENTIAL ‘e’

We have seen that the simplest exponential functions are of the form f(x)¡=¡bx where 6 ¡1: b¡>¡0, b¡= Below are some graphs of simple exponential functions. y=(0.2) x

y=5 x y

y=(0.5)

y=2 x

x

y=1.2 x

1 x

We can see that for all positive values of the base b, the graph is always positive.

bx > 0 for all b > 0.

Hence

There are a vast number of possible choices for the base number. However, where exponential data is examined in science, engineering, and other areas, the base e ¼ 2:7183 is commonly used.

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e is a special number in mathematics. It is irrational like ¼, and just as ¼ is the ratio of a circle’s area to its diameter, e also has a physical meaning. We explore this meaning in the following investigation.

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INVESTIGATION 2

CONTINUOUS COMPOUND INTEREST

A formula for calculating the amount to which an investment grows is un = u0 (1 + i)n where u0 is the initial amount un is the final amount i is the interest rate per compounding period n is the number of periods or number of times the interest is compounded. We will investigate the final value of an investment for various values of n, and allow n to get extremely large. What to do:

1 Suppose $1000 is invested for one year at a fixed rate of 6% p.a. Use your calculator to find the final amount or maturing value if the interest is paid: a annually (n = 1, i = 6% = 0:06) c monthly d daily

b quarterly (n = 4, i = 6% 4 = 0:015) e by the second f by the millisecond.

2 Comment on your answers obtained in 1. 3 If r is the percentage rate per year,

t is the number of years, and r N is the number of interest payments per year, then i = and n = Nt. N ³ r ´Nt This means that the growth formula becomes un = u0 1 + N ·µ ¶a ¸ rt N 1 If we let a = , show that un = u0 1 + : r a

4 For continuous compound growth, the number of interest payments per year N gets very large. a Explain why a gets very large as N gets very large. b Copy and complete the table: Give answers as accurately as technology permits. 5 You should have discovered that for very large a values, µ ¶a 1 1+ ¼ 2:718 281 828 235:::::: a

a

µ ¶a 1 1+ a

10 100 1000 10 000 100 000 .. .

6 Now use the e x key of your calculator to find the value of e1 , i.e., press 1 e x

=

ex 1

or

=

. What do you notice?

un = u0 ert

7 For continuous growth,

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where u0 is the initial amount r is the annual percentage rate t is the number of years Use this formula to find the final value if $1000 is invested for 4 years at a fixed rate of 6% p.a., where the interest is calculated continuously.

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From Investigation 2 we observe that: “If interest is paid continuously (instantaneously) then the formula for calculating a compounding amount un = u0 (1 + i)n can be replaced by un = u0 ert , where r is the percentage rate p.a. and t is the number of years.”

RESEARCH

RESEARCHING e y

What to do:

1 The ‘bell curve’ which models statistical distributions is shown alongside. Research the equation of this curve. i¼

2 e + 1 = 0 is called Euler’s equation where i is the imaginary number Research the significance of this equation. 3 The series f (x) = 1 + x + 12 x2 +

1 3 2£3 x x

+

1 4 2£3£4 x

p

x

¡1.

+ :::::: has infinitely many

terms. It has been shown that f(x) = e . Check this statement by finding an approximation for f (1) using its first 20 terms.

EXERCISE 3H e x key on a calculator to find the value of e to as many digits as possible.

1 Use the

2 Sketch, on the same set of axes, the graphs of y = 2x , y = ex and y = 3x . Comment on any observations.

GRAPHING PACKAGE

3 Sketch, on the same set of axes, the graphs of y = ex and y = e¡x . What is the geometric connection between these two graphs? 4 For the general exponential function y = aekx , what is the y-intercept? 5 Consider y = 2ex . a Explain why y can never be < 0.

b Find y if: i

6 Find, to 3 significant figures, the value of: a e2 b e3 c e0:7

d

7 Write the following as powers of e: p p a e b e e

x = ¡20 ii

p e

e

e¡1

d

1 e2

x = 20.

c

1 p e

c

(e¡0:04 ) 8

d

(e¡0:836 ) 5

e4:829 1000e1:2642

d h

e¡4:829 0:25e¡3:6742

8 Simplify: t

(e0:36 ) 2

a

t

(e0:064 ) 16

b

9 Find, to five significant figures, the values of: b e¡2:31 c a e2:31 ¡0:1764 ¡0:6342 e 50e f 80e g

t

t

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10 On the same set of axes, sketch and clearly label the graphs of: g : x 7! ex¡2 , h : x 7! ex + 3 f : x 7! ex , State the domain and range of each function.

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11 On the same set of axes, sketch and clearly label the graphs of: g : x 7! ¡ex , h : x 7! 10 ¡ ex f : x 7! ex , State the domain and range of each function. t

12 The weight of bacteria in a culture is given by W (t) = 2e 2 grams where t is the time in hours after the culture was set to grow. a What is the weight of the culture at: i t=0 ii t = 30 min

t = 1 12 hours

iii t 2

iv

t = 6 hours?

b Use a to sketch the graph of W (t) = 2e . 13 The current flowing in an electrical circuit t seconds after it is switched off is given by I(t) = 75e¡0:15t amps. a What current is still flowing in the circuit after: i t = 1 sec ii t = 10 sec? b Use your graphics calculator to sketch ¡0:15t and I = 1. I(t) = 75e c Find how long it would take for the current to fall to 1 amp. a Given f : x 7! ex , find the defining equation of f ¡1 . b Sketch the graphs of y = ex , y = x and y = f ¡1 (x) on the same set of axes.

14

REVIEW SET 3A 1 Simplify:

a

¡(¡1)10

b

¡(¡3)3

30 ¡ 3¡1

c

2 Simplify using the index laws: c

5(x2 y)2 (5x2 )2

3 Write the following as a power of 2: b 16 ¥ 2¡3 a 2 £ 2¡4

c

84

4 Write without brackets or negative indices: b (ab)¡1 a b¡3

c

ab¡1

a

a4 b5 £ a2 b2

6xy5 ¥ 9x2 y 5

b

2x¡3 =

5 Find the value of x, without using your calculator: a 6 Evaluate without using a calculator:

a

2

83

b

1 32

b

9x = 272¡2x

¡2 3

27

7 Evaluate, correct to 3 significant figures, using your calculator: a

3

34

¡1 5

b

c

27

8 If f(x) = 3 £ 2x , find the value of:

a f (0)

b f(3)

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9 On the same set of axes draw the graphs of a y = 2x b the y-intercept and the equation of the horizontal asymptote.

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f (¡2)

y = 2x ¡ 4, stating

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10 The temperature of a liquid t minutes after it was heated is given by T = 80 £ (0:913)t o C. Find: a the initial temperature of the liquid b the temperature after i t = 12 ii t = 24 iii t = 36 minutes. c Draw the graph of T against t for t > 0, using the above or technology. d Hence, find the time taken for the temperature to reach 25o C.

REVIEW SET 3B 1 Simplify:

¡(¡2)3

a

5¡1 ¡ 50

b

2 Simplify using the index laws: a

(a7 )3

pq 2 £ p3 q 4

b

3 Write as powers of 2:

1 16

a

2x £ 4

b

c

4 Write without brackets or negative indices: b 2(ab)¡2 a x¡2 £ x¡3 5 Solve for x without using a calculator: 6 Write as powers of 3:

a

7 Write as a single power of 3:

8 For y = 3x ¡ 5 : a find y when x = 0, §1, §2 c sketch the graph of y = 3x ¡ 5

2x+1 = 32

2ab¡2 b

d

4x+1 =

¡ 1 ¢x 8

1 243

p ( 3)1¡x £ 91¡2x

b

discuss y as x ! 1 and as x ! ¡1 state the equation of any asymptote.

b d

9 Without using a calculator, solve for x:

1 27

c

1

27 9a

a

4x ¥ 8 c

a

b

81

8ab5 2a4 b4

c

27x = 3

a

b

91¡x = 27x+2

10 On the same set of axes, sketch and clearly label the graphs of: f : x 7! ex , g : x 7! ex¡1 , h : x 7! 3 ¡ ex State the domain and range of each function.

REVIEW SET 3C a Write 4 £ 2n as a power of 2. b Evaluate 7¡1 ¡ 70 . c Write ( 23 )¡3 in simplest fractional form. µ ¡1 ¶2 2a d Simplify . Do not have negative indices or brackets in your answer. b2 2x+1 . 2 a Write 288 as a product of prime numbers in index form. b Simplify 21¡x 3 Write as powers of 5 in simplest form: p 1 a 1 b 5 5 c p d 25a+3 4 5

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4 Simplify: a ¡(¡2)2

b

(¡ 12 a¡3 )2

c

(¡3b¡1 )¡3

5 Expand and simplify: a ex (e¡x + ex )

b

(2x + 5)2

c

(x 2 ¡ 7)(x 2 + 7)

6 Expand and simplify: a (3 ¡ 2a )2

b

p p ( x + 2)( x ¡ 2)

c

2¡x (22x + 2x )

1

b 4 £ ( 13 )x = 324

a 6 £ 2x = 192

7 Solve for x:

1

8 The weight of a radioactive substance after t years is given by W = 1500 £ (0:993)t grams. a Find the original amount of radioactive material. b Find the amount of radioactive material remaining after: i 400 years ii 800 years. c Sketch the graph of W against t, t > 0, using the above or technology. d Hence, find the time taken for the weight to reduce to 100 grams.

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b discuss y as x ! 1 and as x ! ¡1 d state the equation of any asymptote.

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10 For y = 2e¡x + 1 : a find y when x = 0, §1, §2 c sketch the graph of y = 2e¡x + 1

16a . 2b

25

8a 1 = , find b 4 128

5

9 If 4a 2b = 1 and

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Chapter

Logarithms Contents:

A B C D E F G H

4

Logarithms Logarithms in base 10 Laws of logarithms Natural logarithms Exponential equations using logarithms The change of base rule Graphs of logarithmic functions Growth and decay

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Review set 4A Review set 4B Review set 4C Review set 4D

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LOGARITHMS (Chapter 4)

OPENING PROBLEM Paulo knows that when he invests E12 000 for n years at an interest rate of 8:35% p.a. compounded annually, the value of the investment at the end of this period is given by An+1 = 12 000 £ (1:0835)n euros. Consider the following questions: 1 What is the value of A1 and what is its interpretation? 2 How would we find the value of the investment after 5 years? 3 If we let n = 2:25, A3:25 = 12 000 £ (1:0835)2:25 : Does the power 2:25 have a meaning? What is the interpretation of the value of A3:25 ? 4 How long would it take for the investment to double in value? 5 What would the graph of An+1 against n look like? After studying the concepts of this chapter, you should be able to answer the questions above.

A

LOGARITHMS

HISTORICAL NOTE In the late 16th century, astronomers spent a large part of their working lives doing the complex and tedious calculations of spherical trigonometry needed to understand the movement of celestial bodies. A Scotsman, John Napier, discovered a method of simplifying these calculations using logarithms. So effective was Napier’s method that it was said he effectively doubled the life of an astronomer by reducing the time required to do these calculations. Consider the function f : x 7! 10x . f ( x) = 10 x or y = 10 x

The defining equation of f is f (x) = 10x or y = 10x .

y

y=x

Now consider the graph of f and its inverse function f ¡1 . f -1

The question arises:

1 ¡1

How can we write f in functional form, or, what is the defining function of f ¡1 ?

1

x

x

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As f is defined by y = 10 , f ¡1 is defined by x = 10y . finterchanging x and yg

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So,

y is the exponent to which 10 (the base) is raised in order to get x. We write this as y = log10 x and say that

So,

“y is the logarithm of x in base 10.”

²

if f (x) = 10x ,

then

f ¡1 (x) = log10 x

²

if f (x) = bx ,

then

f ¡1 (x) = logb x

LOGARITHMS IN BASE b If A = bn , b 6= 1, b > 0, we say that n is the logarithm of A in base b, and that A = bn , n = logb A, A > 0.

In general:

A = bn , n = logb A is a short way of writing: “if A = bn then n = logb A, and if n = logb A then A = bn ”. We say that A = bn

8 = 23 means that 3 = log2 8 and vice versa. log5 25 = 2 means that 25 = 52 and vice versa.

² ²

For example:

and n = logb A are equivalent or interchangeable statements.

If y = bx

then x = logb y, and so

x = logb bx .

If x = by

then y = logb x, and so

x = blogb x

provided x > 0.

Example 1 a b

Write an equivalent exponential statement for log10 1000 = 3. Write an equivalent logarithmic statement for 34 = 81.

a

From log10 1000 = 3 we deduce that 103 = 1000.

b

From 34 = 81 we deduce that log3 81 = 4.

Example 2 Find: a log10 100

b log2 32

c

log5 (0:2)

a

To find log10 100 we ask “What power must 10 be raised to, to get 100?” As 102 = 100, then log10 100 = 2.

b

As 25 = 32, then log2 32 = 5.

c

As 5¡1 =

1 5

= 0:2, then log5 (0:2) = ¡1.

EXERCISE 4A 1 Write an equivalent exponential statement for: b log10 (0:1) = ¡1 a log10 10 000 = 4

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log2 ( 14 ) = ¡2

5

e

log2 8 = 3

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Y:\HAESE\IB_HL-2ed\IB_HL-2ed_04\103IB_HL-2_04.CDR Friday, 4 January 2008 9:07:37 AM DAVID3

c f

p log10 10 = 12 p log3 27 = 1:5

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2 Write an equivalent logarithmic statement for: b 2¡3 = 18 a 22 = 4 e 26 = 64 d 72 = 49 3 Find: a log10 100 000 e log2 64 i log2 (0:125) m

b f j

q

r

4 Use your calculator to find: b log10 25 a log10 152 5 Solve for x: a log2 x = 3

b

log4 x =

1 2

10¡2 = 0:01 1 3¡3 = 27

c g k

p log3 3 log5 25 log4 16

d h l

log2 8 log5 125 log36 6

o

loga an

p

log8 2

s

log4 1

t

log9 9

c

log10 74

log10 (0:01) log2 128 log9 3 p log2 3 2 p log6 6 6

n

log3 243 ¡ ¢ logt 1t

c f

c

d

d

logx 81 = 4

log10 0:8

log2 (x ¡ 6) = 3

a Prove that loga an = n.

6

b Hence, find: i log4 16 ¶ µ 1 v log2 p 2

ii

log2 4

iii

log3

vi

p log5 (25 5)

vii

log3

B

¡1¢ 3

³

p1 3

iv ´ viii

p log10 5 100 ¶ µ 1 p log4 2 2

LOGARITHMS IN BASE 10 10 000 = 104 1000 = 103 100 = 102 10 = 101 1 = 100 0:1 = 10¡1 0:01 = 10¡2 0:001 = 10¡3

For example,

Many positive numbers can be easily written in the form 10x .

etc. p p 1 Numbers like 10, 10 10 and p can also be written in the form 10x as follows: 5 10 p p 1 p 10 10 10 5 10 1 = 101 £ 100:5 1 = 10 2 = 10¡ 5 1:5 = 10 0:5 = 10 = 10¡0:2 In fact, all positive numbers can be written in the form 10x by using logarithms in base 10.

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The logarithm in base 10 of a positive number is its power of 10.

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For example: ²

Since 1000 = 103 , we write log10 1000 = 3 or log 1000 = 3.

²

Since 0:01 = 10¡2 , we write log10 (0:01) = ¡2 or log(0:01) = ¡2. a = 10 log a

In algebraic form,

for any a > 0.

If no base is indicated we assume it means base 10.

Notice that a must be positive since 10 > 0 for all x 2 R . x

Notice also that

log 1000 = log 103 = 3

and

log 0:01 = log 10¡2 = ¡2 log 10 x = x

give us the useful alternative

Example 3 a b

Without using a calculator, find: i log 100 Check your answers using technology.

a

i log 100 = log 102 = 2

b

i

press

log

100

ii

press

log

10 ^ 0:25 )

)

ii

p log( 4 10):

p 1 log( 4 10) = log(10 4 ) =

ii

1 4

Answer: 2

ENTER

Answer: 0:25

ENTER

Example 4 Use your calculator to write the following in the form 10x where x is correct to 4 decimal places: a 8 b 800 c 0:08

a

b

8 = 10log 8 ¼ 100:9031

800 = 10log 800 ¼ 102:9031

c

0:08 = 10log 0:08 ¼ 10¡1:0969

Example 5 Use your calculator to find: i Explain why log 20 = log 2 + 1:

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log 2

log 20

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b log 20 = log(2 £ 10) ¼ log(100:3010 £ 101 ) fadding indicesg ¼ log 101:3010 ¼ 1:3010 ¼ log 2 + 1

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log 20 ¼ 1:3010 fcalculatorg

0

ii

5

log 2 ¼ 0:3010

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Example 6 x = 10log x x = 103 x = 1000

a

Find x if: a log x = 3 b log x ¼ ¡0:271

) )

x = 10log x x ¼ 10¡0:271 x ¼ 0:536

b ) )

EXERCISE 4B 1 Without using a calculator, a log 10 000 b p f e log 10 p j i log 3 100 log 10n

m

n

find: log 0:001 p log( 3 10) ³ ´ 100 log p 10

c g k

log(10a £ 100)

o

2 Find using a calculator: a e

b

log 10 000 p log 3 100

f

p log 10 ³ ´ log p110

c

log 0:001 p log 10 10

log 10 ³ ´ 1 log p 4 10 p log(10 £ 3 10) µ ¶ 10 log 10m

g

d h l p

log 1

p log 10 10 p log 1000 10 µ a¶ 10 log 10b

d

log

h

log

p 3 10 ³ ´ 1 p 4 10

3 Use your calculator to write these in the form 10x where x is correct to 4 decimal places: a 6 b 60 c 6000 d 0:6 e 0:006 f 15 g 1500 h 1:5 i 0:15 j 0:000 15 4

a Use your calculator to find: i log 3 b Explain why log 300 = log 3 + 2:

ii

log 300

5

a Use your calculator to find: i log 5 b Explain why log 0:05 = log 5 ¡ 2:

ii

log 0:05

6 Find x if: a log x = 2 e log x = 12

b f

c g

log x = 1 log x = ¡ 12

C

log x = 0 log x ¼ 0:8351

d h

log x = ¡1 log x ¼ ¡3:1997

LAWS OF LOGARITHMS

INVESTIGATION

DISCOVERING THE LAWS OF LOGARITHMS

What to do:

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1 Use your calculator to find: a log 2 + log 3 b log 3 + log 7 c log 4 + log 20 d log 6 e log 21 f log 80 From your answers, suggest a possible simplification for ln a + ln b.

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2 Use your calculator to find: a log 6 ¡ log 2 b log 12 ¡ log 3 c log 3 ¡ log 5 d log 3 e log 4 f log(0:6) From your answers, suggest a possible simplification for ln a ¡ ln b: 3 Use your calculator to find: a 3 log 2 b 2 log 5 c ¡4 log 3 e log(52 ) f log(3¡4 ) d log(23 ) From your answers, suggest a possible simplification for n log a. From the investigation, you should have discovered the three important laws of logarithms: ²

If A and B are both positive then:

² ²

log A + log B = log(AB) µ ¶ A log A ¡ log B = log B n log A = log (An )

More generally, in any base c we have these laws of logarithms: ²

If A and B are both positive then:

² ²

logc A + logc B = logc (AB) µ ¶ A logc A ¡ logc B = logc B n logc A = logc (An )

These laws are easily established using the first three index laws.

Example 7 Use the laws of logarithms to write the following as a single logarithm: a log 5 + log 3 b log 24 ¡ log 8 c log 5 ¡ 1 a

log 24 ¡ log 8 ¡ ¢ = log 24 8 = log 3

b

log 5 + log 3 = log(5 £ 3) = log 15

c

log 5 ¡ 1 = log 5 ¡ log 101 ¡5¢ = log 10 ¡ ¢ = log 12

Example 8

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2 log 7 ¡ 3 log 2 = log(72 ) ¡ log(23 ) = log 49 ¡ log 8 ¡ ¢ = log 49 8

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a

Write as a single logarithm in the form log a, a 2 Q . a 2 log 7 ¡ 3 log 2 b 2 log 3 ¡ 1

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b

2 log 3 ¡ 1 = log(32 ) ¡ log 101 = log 9 ¡ log 10 = log(0:9)

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Example 9 log 8 log 4

Simplify

log 8 log 23 = log 4 log 22 3 log 2 = 2 log 2 = 32

without

using a calculator.

Example 10

a

log

Show that: ¡ ¢ a log 19 = ¡2 log 3

¡1¢

b

9

log 500 ¢ ¡ = log 1000 2

= log(3¡2 ) = ¡2 log 3

= log 1000 ¡ log 2 = log 103 ¡ log 2 = 3 ¡ log 2

b log 500 = 3 ¡ log 2

EXERCISE 4C.1 1 Write a d g j m

as a single logarithm: log 8 + log 2 log 4 + log 5 1 + log 3 2 + log 2

b e h k

log 8 ¡ log 2 log 5 + log(0:4) log 4 ¡ 1 log 40 ¡ 2

c f i l

log 50 ¡ 4

n

3 ¡ log 50

o

2 Write as a single logarithm or integer: a 5 log 2 + log 3 b 2 log 3 + 3 log 2 2 log 5 ¡ 3 log 2

e

1 2

g

3 ¡ log 2 ¡ 2 log 5

h

1 ¡ 3 log 2 + log 20

log 4 + log 3

3 Simplify without using a calculator: log 27 log 4 a b log 2 log 9 log 3 log 9

d

log 25 log(0:2)

e

f

3 log 4 ¡ log 8 ¡1¢ 1 3 log 8

i

2¡

c

log 8 log 2

f

log 8 log(0:25)

c

d

log 40 ¡ log 5 log 2 + log 3 + log 4 log 5 + log 4 ¡ log 2 log 6 ¡ log 2 ¡ log 3 ¡ ¢ log 43 + log 3 + log 7

1 2

log 4 ¡ log 5

Check your answers using a calculator. 4 Show that: a log 9 = 2 log 3 ¡ ¢ d log 15 = ¡ log 5

b

p log 2 =

log 2

c

log

e

log 5 = 1 ¡ log 2

f

log 5000 = 4 ¡ log 2

1 2

¡1¢ 8

= ¡3 log 2

5 If p = logb 2, q = logb 3 and r = logb 5 write in terms of p, q and r:

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logb 45

f

logb (0:2)

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logb 108 ¡5¢ logb 32

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d

b

logb 6 ³ p ´ logb 5 2 3

5

a

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6 If log2 P = x, log2 Q = y a

log2 (P R)

d

p log2 (P Q)

and log2 R = z b

log2 (RQ2 ) µ

2

e

write in terms of x, y and z: µ ¶ PR c log2 Q µ 2p ¶ R Q f log2 P3

Q3 p R

log2

¶

7 If logt M = 1:29 and logt N 2 = 1:72 find: a

b

logt N

µ c

logt (MN)

logt

N2 p M

¶

LOGARITHMIC EQUATIONS Example 11 a y = a2 b

Write these as logarithmic equations (in base 10): a ) ) )

c

y log y log y log y

= a2 b = log(a2 b) = log a2 + log b = 2 log a + log b

µ

¶

P =

20 p n

b y=

20 1 n2

20 P =p n

y=

)

µ

c

a b3 ³ ´ a log y = log 3 b log y = log a ¡ log b3 log y = log a ¡ 3 log b

b

) log P = log

a b3

¶

) )

and so log P = log 20 ¡

1 2

log n

Example 12 a log A = log b + 2 log c b log M = 3 log a ¡ 1

Write the following equations without logarithms:

a

log M = 3 log a ¡ 1 ) log M = log a3 ¡ log 101 µ 3¶ a ) log M = log 10 a3 ) M= 10

b

log A = log b + 2 log c ) log A = log b + log c2 ) log A = log(bc2 ) ) A = bc2

EXERCISE 4C.2

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M = ad4

d

g

y = abx

h

k

S = 200 £ 2t

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0

ab c

f

5

L=

y = 20b3 a Q= n b r a N= b

b

100

i

a

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e

y = 2x p R=b l

75

1 Write the following as logarithmic equations (in base 10):

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p T =5 d 20 F =p n m a y= n b

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2 Write the following equations without logarithms: a log D = log e + log 2 b log F = log 5 ¡ log t 1 c log P = 2 log x d log M = 2 log b + log c e log B = 3 log m ¡ 2 log n f log N = ¡ 13 log p g log P = 3 log x + 1 h log Q = 2 ¡ log x 3 Solve for x: a log3 27 + log3 ( 13 ) = log3 x p c log5 125 ¡ log5 5 = log5 x e log x + log(x + 1) = log 30

D

b

log5 x = log5 8 ¡ log5 (6 ¡ x)

d f

log20 x = 1 + log20 10 log(x + 2) ¡ log(x ¡ 2) = log 5

NATURAL LOGARITHMS

In Chapter 3 we came across the natural exponential e ¼ 2:718 28 . If f is the exponential function x 7! ex (i.e., f(x) = ex or y = ex ) then its inverse function, f ¡1 is x = ey or y = loge x. y = loge x is the reflection of y = ex in the mirror line y = x. ln x is used to represent loge x. ln x is called the natural logarithm of x.

So,

ln 1 = ln e0 = 0

Notice that:

p 1 ln e = ln e 2 =

ln e = ln e1 = 1 ln e2 = 2 µ ¶ 1 and ln = ln e¡1 = ¡1 e

1 2

ln ex = x and eln x = x.

In general,

¢x ¡ Also, since ax = eln a = ex ln a ,

ax = ex ln a .

Example 13 Use your calculator to write the following in the form ek where k is correct to 4 decimal places: a 50 b 0:005

a

b

50 = eln 50 ¼ e3:9120

ln x

fusing x = e

0:005 = eln 0:005 ¼ e¡5:2983

g

Example 14

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ln x = 2:17 ) x = e2:17 ) x ¼ 8:76

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a

Find x if: a ln x = 2:17 b ln x = ¡0:384

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ln x = ¡0:384 ) x = e¡0:384 ) x ¼ 0:681

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Example 15 Use the laws of logarithms to write the following as a single logarithm: a ln 5 + ln 3 b ln 24 ¡ ln 8 c ln 5 ¡ 1 a

ln 24 ¡ ln 8 ¡ ¢ = ln 24 8 = ln 3

b

ln 5 + ln 3 = ln(5 £ 3) = ln 15

ln 5 ¡ 1 = ln 5 ¡ ln e1 ¡ ¢ = ln 5e

c

Example 16 2 ln 7 ¡ 3 ln 2 = ln(72 ) ¡ ln(23 ) = ln 49 ¡ ln 8 ¡ ¢ = ln 49 8

a Use the laws of logarithms to simplify: a 2 ln 7 ¡ 3 ln 2 b 2 ln 3 ¡ 1

Example 17 a

Show that: ¡ ¢ a ln 19 = ¡2 ln 3

ln

b

µ

¡1¢

b

9

ln 500 = ln

= ln(3¡2 ) = ¡2 ln 3

b ln 500 ¼ 6:9078 ¡ ln 2

2 ln 3 ¡ 1 = ln(32 ) ¡ ln e = ln 9 ¡ ln e µ ¶ 9 = ln e

1000 2

¶

= ln 1000 ¡ ln 2 ¼ 6:9078 ¡ ln 2

Example 18 Write the following equations without logarithms: a ln A = 2 ln c + 3 b ln M = 3 ln a ¡ 2 a

ln M = 3 ln a ¡ 2 ) ln M ¡ 3 ln a = ¡2 ) ln M ¡ ln a3 = ¡2 µ ¶ M ) ln = ¡2 a3

b

ln A = 2 ln c + 3 ) ln A ¡ 2 ln c = 3 ) ln A ¡ ln c2 = 3 µ ¶ A ) ln 2 = 3 c

M = e¡2 a3 ) M = a3 e¡2

A = e3 c2 ) A = e3 c2

)

)

or M =

a3 e2

EXERCISE 4D 1 Without using a calculator find: b ln 1 a ln e3

c

p ln 3 e

µ d

ln

1 e2

¶

2 Check your answers to question 1 using a calculator.

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3 Explain why ln(¡2) and ln 0 cannot be found.

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µ

¶ ea d ln(e ) b ln(e £ e ) c e ln eb x 5 Use your calculator to write these in the form e where x is correct to 4 dec. places: a 6 b 60 c 6000 d 0:6 e 0:006 f 15 g 1500 h 1:5 i 0:15 j 0:000 15

4 Simplify: a ln ea

¢ ¡ ln ea £ eb

a

6 Find x if: a ln x = 3 e ln x = ¡5

b f

c g

ln x = 1 ln x ¼ 0:835

7 Write as a single logarithm: a ln 15 + ln 3 d ln 4 + ln 6 g 1 + ln 4 j 2 + ln 4

a b

ln x = ¡1 ln x ¼ ¡3:2971

d h

ln x = 0 ln x ¼ 2:145

ln 15 ¡ ln 3 ln 5 + ln(0:2) ln 6 ¡ 1 ln 20 ¡ 2

c f i l

ln 20 ¡ ln 5 ln 2 + ln 3 + ln 5 ln 5 + ln 8 ¡ ln 2 ln 12 ¡ ln 4 ¡ ln 3

8 Write in the form ln a, a 2 Q : a 5 ln 3 + ln 4 b 3 ln 2 + 2 ln 5

c

i

3 ln 2 ¡ ln 8 ¡1¢ 1 3 ln 27 ¡ ¢ ¡2 ln 14

c

ln

f

b e h k

d

3 ln 4 ¡ 2 ln 2

e

g

¡ ln 2

h

9 Show that: a ln 27 = 3 ln 3 ¡ ¢ d ln 16 = ¡ ln 6

b

1 3

f

ln 8 + ln 3 ¡ ¢ ¡ ln 12 p ln 3 = 12 ln 3 ³ ´ ln p12 = ¡ 12 ln 2

¡

1 16

¢

³e´

= ¡4 ln 2

= ¡5 ln 2

i

= 1 ¡ ln 5 ´ 1 ln p = ¡ 15 ln 2 5 2

10 Write the following equations without logarithms: a ln D = ln x + 1 b ln F = ¡ ln p + 2

c

ln P =

f

ln N = ¡ 13 ln g

g

ln

p 3 5=

1 3

e h

ln 5

ln

¡

1 32

¢

³

d

ln M = 2 ln y + 3

e

ln B = 3 ln t ¡ 1

g

ln Q ¼ 3 ln x + 2:159

h

ln D ¼ 0:4 ln n ¡ 0:6582

E

ln

5

1 2

ln x

EXPONENTIAL EQUATIONS USING LOGARITHMS

In Chapter 3 we found solutions to simple exponental equations by creating equal bases and then equating indices. However, it is not always easy to make the bases the same. In these situations we use logarithms to find the solution.

Example 19 Solve for x, giving your answer to 3 significant figures: 2x = 30.

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ffind the logarithm of each sideg

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2x = 30 log 2x = log 30

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)

flog an = n log ag

x log 2 = log 30 log 30 ) x= log 2 ) x ¼ 4:91 (3 s.f.)

Example 20 Solve for t to 3 significant figures: 200 £ 20:04t = 6: 200 £ 20:04t = 6 6 ) 20:04t = 200

fdividing both sides by 200g

0:04t

) 2 = 0:03 0:04t = log 0:03 ffind the logarithm of each sideg ) log 2 ) 0:04t £ log 2 = log 0:03 flog an = n log ag log 0:03 ) t= ¼ ¡126 (3 s.f.) 0:04 £ log 2

EXERCISE 4E 1 Solve for x, giving your answer correct to 3 significant figures: b 3x = 20 c 4x = 100 a 2x = 10 e 2x = 0:08 f 3x = 0:000 25 d (1:2)x = 1000 ( 12 )x = 0:005

g

h

( 34 )x = 10¡4

i

(0:99)x = 0:000 01

2 Find the solution to the following correct to 4 significant figures: a

200 £ 20:25t = 600

b

20 £ 20:06t = 450

c

30 £ 3¡0:25t = 3

d

12 £ 2¡0:05t = 0:12

e

50 £ 5¡0:02t = 1

f

300 £ 20:005t = 1000

To solve exponential equations of the form ex = a we simply use the property:

If ex = a then x = ln a.

This rule is clearly true, because if ex = a then ln ex = ln a ) x = ln a

ffinding ln of both sidesg fln ex = xg

Example 21 x

b e 3 = 21:879

Find x to 4 s.f. if: a ex = 30 x

e 3 = 21:879 x ) = ln 21:879 3

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ex = 30 ) x = ln 30 ) x ¼ 3:401

25

a

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c

20e4x = 0:0382

c

20e4x = 0:0382 ) e4x = 0:001 91 ) 4x = ln 0:001 91 ) 4x ¼ ¡6:2607 ) x ¼ ¡1:565

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3 Solve for x, giving answers correct to 4 significant figures: b ex = 1000 a ex = 10 x

e2 = 5 20 £ e0:06x = 8:312

d g

x

e 3 = 157:8 50 £ e¡0:03x = 0:816

e h

c

ex = 0:008 62

f i

e 10 = 0:016 82 41:83e0:652x = 1000

x

Note:

Remember that you may not need to use logs when solving exponential equations. It is usually much easier if you can get the same base on both sides.

F

THE CHANGE OF BASE RULE

If logb A = x,

then bx = A ) logc bx = logc A ) x logc b = logc A logc A ) x= logc b

ftaking logarithms in base cg fpower law of logarithmsg logb A =

So,

logc A logc b

Example 22 a letting log2 9 = x logc A b using the rule logb A = logc b i c = 10 ii c = e

Find log2 9 by:

a

Let log2 9 = x ) 9 = 2x ) log 2x = log 9 ) x log 2 = log 9 log 9 ¼ 3:17 ) x= log 2

with:

log10 9 log10 2 ¼ 3:17

b i

log2 9 =

ii

log2 9 =

ln 9 ln 2 ¼ 3:17

Example 23 Solve for x: 8x ¡ 5(4x ) = 0

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8x ¡ 5(4x ) = 0 ) 23x ¡ 5(22x ) = 0 ) 22x (2x ¡ 5) = 0 fas 22x > 0 for all xg ) 2x = 5 ) x = log2 5 log 5 ¼ 2:32 fCheck this using technology.g ) x= log 2

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EXERCISE 4F log10 A log10 b

1 Use the rule logb A = a

b

log3 12

2x = 0:051

a

c

log 12 1250 ln A ln b

2 Use the rule logb A =

to find, correct to 3 significant figures: log0:4 (0:006 984)

to solve, correct to 3 significant figures:

4x = 213:8

b

d

log3 (0:067)

c

32x+1 = 4:069

Hint: In 2a 2x = 0:051 implies that x = log2 (0:051) . 3 Solve for x: a 25x ¡ 3(5x ) = 0

b

8(9x ) ¡ 3x = 0

4 Solve for x: p a log4 x3 + log2 x = 8

b

log16 x5 = log64 125 ¡ log4

p x

5 Find the exact value of x for which 4x £ 54x+3 = 102x+3

G GRAPHS OF LOGARITHMIC FUNCTIONS Consider the general exponential function f : x 7! ax , a > 0, a 6= 1: The defining equation of f is f (x) = ax (or y = ax ). The graph of y = ax is: For 0 < a < 1:

For a > 1: y

y

(-1, 1 ) a

(-1, 1 ) a

(1, a)

1

1

(1, a) x

x

These functions have the horizontal asymptote y = 0 (the x-axis). They have domain R (all real numbers) and range fy : y > 0g or y 2 ] 0, 1 [. Obviously the function y = ax

is one-to-one and has an inverse function f ¡1 .

So, if f is y = ax , then f ¡1 is x = ay , i.e., y = loga x. If f (x) = ax

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The domain of f = the range of f ¡1 . The range of f = the domain of f ¡1 .

5

²

95

The domain of f ¡1 is fx : x > 0g or x 2 ] 0, 1 [. The range of f ¡1 is y 2 R .

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Note:

then f ¡ 1 (x) = loga x.

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LOGARITHMIC GRAPHS The graphs of y = loga x are: For 0 < a < 1:

For a > 1:

y¡=¡a x

y¡=¡a x

y

y

y = log a x

(a, 1)

y = log a x

(1a , - 1)

x

1

x

1

y¡=¡x

(a, 1)

1

1

( 1a , - 1)

y=x

Note: ² ² ² ² ²

both graphs are reflections of y = ax in the line y = x both functions have domain fx: x > 0g or x 2 ] 0, 1 [ we can only find logarithms of positive numbers both graphs have the vertical asymptote x = 0 (the y-axis) for 0 < a < 1, as x ! 1, y ! ¡1 and as x ! 0 (from right), y ! 1 for a > 1, as x ! 1, y ! 1 and as x ! 0 (from right), y ! ¡1 ² to find the domain of loga g(x), we find the solutions of g(x) > 0.

Example 24 Consider the function f : x 7! log2 (x ¡ 1) + 1: a Find the domain and range of f . b Find any asymptotes and axis intercepts. c Sketch the graph of f showing all important features. d Find f ¡1 and explain how to verify your answer. a

x ¡ 1 > 0 when x > 1

So, the domain is x 2 ] 1, 1 [ and the range is y2R

b

As x ! 1 from the right, y ! ¡1. ) x = 1 is the vertical asymptote. As x ! 1, y ! 1: When x = 0, y is undefined ) there is no y-intercept. ) x ¡ 1 = 2¡1 ) x = 1 12 When y = 0, log2 (x ¡ 1) = ¡1 So, the x-intercept is 1 12

c

6

f is defined by y = log2 (x ¡ 1) + 1 ) f is defined by x = log2 (y ¡ 1) + 1 ) x ¡ 1 = log2 (y ¡ 1)

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To graph using your calculator we will need to change the base. log(x ¡ 1) So, we graph y = +1 log 2

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)

y ¡ 1 = 2x¡1 ) y = 2x¡1 + 1 ) f ¡1 (x) = 2x¡1 + 1 which has a H.A. of y = 1 X Its domain is x 2 R , range is y 2 ] 1, 1 [ . Graphics calculator tip: When graphing f, f ¡1 and y = x on the same axes, it is best to set the scale so that y = x makes a 45o angle with both axes. Why? To ensure the graphs are not distorted, use a square window. Recall that:

² ²

inverse functions are formed by interchanging x and y y = f ¡1 (x) is the reflection of y = f (x) in the line y = x.

Example 25 Given f : x 7! ex¡3 a find the defining equation of f ¡1 b sketch the graphs of f and f ¡1 on the same set of axes c state the domain and range of f and f ¡1 d find any asymptotes and intercepts. f (x) = ex¡3 ) f is x = ey¡3 ) y ¡ 3 = ln x ) y = 3 + ln x

a

b

y

¡1

c domain range d

y = 3 + ln x

(1, 3)

f ¡1 x>0 y2R

f x2R y>0

y = e x -3

(3, 1) x y=x

For f: HA is y = 0, for f ¡1 : VA is x = 0. ¡3 For f: y-int is (0, e ), for f ¡1 : x-int is (e¡3 , 0).

EXERCISE 4G.1 1 For the following functions f : i Find the domain and range. ii Find any asymptotes and axes intercepts. iii Sketch the graph of y = f (x) showing all important features. iv Solve f(x) = ¡1 algebraically and check the solution on your graph. v Find f ¡1 and explain how to verify your answer.

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f : x 7! log3 (x + 1) f : x 7! log5 (x ¡ 2) ¡ 2 f :x! 7 1 ¡ log2 x2

a c e

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2 For the following functions: i find the defining equation of f ¡1 ii sketch the graphs of f and f ¡1 on the same set of axes iii state the domain and range of f and f ¡1 . iv Find any asymptotes. a c

f : x 7! ex + 5 f : x 7! ln x ¡ 4 where x > 0

3 Given f : x 7! e2x a

(f

¡1

f : x 7! ex+1 ¡ 3 f : x 7! ln(x ¡ 1) + 2 where x > 1

b d

and g : x 7! 2x ¡ 1, find the defining equations of:

± g)(x)

(g ± f )¡1 (x)

b

4 Consider the graphs A and B. One of them is the graph of y = ln x and the other is the graph of y = ln(x ¡ 2): a Identify which is which. Give evidence for your answer. b Redraw the graphs on a new set of axes and add to them the graph of y = ln(x + 2):

y A B 1

3

x

c Find the equation of the vertical asymptote for each graph. 5 Kelly said that in order to graph y = ln(x2 ), you could first graph y = ln x and then double the distances away from the x-axis. Connecting these points will give the graph of y = ln x2 .

y

Is she correct? Give evidence.

y = ln&x2*

1

y = ln x

x

6 For the function f : x 7! ex+3 + 2 a Find the defining equation for f ¡1 . b Find the values of x for which: i f (x) < 2:1 ii f (x) < 2:01 iii f(x) < 2:001 iv f(x) < 2:0001 and hence conjecture the horizontal asymptote for the graph of f . c Determine the horizontal asymptote of f (x) by discussing the behaviour of f (x) as x ! §1: d Hence, determine the vertical asymptote and the domain of f ¡1 .

FURTHER INEQUALITIES We have seen on many occasions what it means to solve an equation. Usually the equations have been presented to us in one of the forms f (x) = 0 or f (x) = g(x):

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Simply, it means we must find all possible values of the pronumeral, x in this case, that make the equation true.

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For equations of the form f (x) = 0, we can graph f (x) and then find where the graph meets the x-axis. For equations of the form f(x) = g(x), we can either graph f (x) and g(x) separately and find the x-coordinate(s) of their point(s) of intersection, or, we can graph y = f (x) ¡ g(x) and find where this graph meets the x-axis. We can use these same principles in order to solve inequalities. Note that when solving inequalities, only real number solutions are possible.

Example 26 a ex = 2x2 + x + 1

Solve for x:

b ex > 2x2 + x + 1

We graph f(x) = ex and g(x) = 2x2 + x + 1:

a

y¡=¡2xX¡+¡x¡+¡1

Using technology we find the points of intersection of f (x) and g(x): ) x = 0 and x ¼ 3:21 are solutions.

» 3.21

We need to be sure that the graphs will not meet again. We could graph f (x) ¡ g(x) and find where the graph meets the x-axis.

Note:

y¡=¡ex

y

y=ƒ(x)-g(x)

y x¡=¡0

x

x¡»¡3.21 x

Using the same graphs as above, we seek values of x for which f (x) > 0. This is where the graph of f (x) either meets or is higher than the graph of g(x). The solution is x = 0 or x > 3:21 0 3 x We could even graph the solution set:

b

»3.21

and could describe it as x = 0 or x 2 [ 3:21, 1 [:

EXERCISE 4G.2 1 Solve for x: a x2 > ex

b

x3 < e¡x

c

5 ¡ x > ln x

2 State the domain of f : x 7! x2 ln x. Hence find where f (x) 6 0: a Use technology to sketch the graph of f : x 7!

3

2 2 ¡ e2x ¡x+1 : x

b State the domain and range of this function.

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c Hence find all x 2 R for which e2x

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H

GROWTH AND DECAY

In Chapter 3 we showed how exponential functions can be used to model a variety of growth and decay situations. These included the growth of populations and the decay of radioactive substances. In this section we consider more growth and decay problems, focussing particularly on how logarithms can be used in their solution.

POPULATION GROWTH Example 27 A farmer monitoring an insect plague notices that the area affected by the insects is given by An = 1000 £ 20:7n hectares, where n is the number of weeks after the initial observation. a Draw an accurate graph of An against n and use your graph to estimate the time taken for the affected area to reach 5000 ha. b Check your answer to a using logarithms and using suitable technology. a

b

An

When An = 5000,

1000 £ 20:7n = 5000 6000 ) 20:7n = 5 5000 ) log 20:7n = log 5 4000 ) 0:7n log 2 = log 5 log 5 ) n= 2000 0:7 £ log 2 ¼3.3 ) n ¼ 3:32 ) it takes about 3 weeks and 0 1 2 3 4 n (weeks) 2 more days. Using technology we find the intersection of y = 1000 £ 20:7x and y = 5000. This confirms n ¼ 3:32 .

EXERCISE 4H.1 1 The weight Wt of bacteria in a culture t hours after establishment is given by Wt = 20 £ 20:15t grams. Find the time for the weight of the culture to reach: a 30 grams b 100 grams. 2 The mass Mt of bacteria in a culture t hours after establishment is given by Mt = 25 £ e0:1t grams. Find the time for the mass of the culture to reach: a 50 grams b 100 grams.

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3 A biologist monitoring a fire ant infestation notices that the area affected by the ants is given by An = 2000 £ e0:57n hectares, where n is the number of weeks after the initial observation. a Draw an accurate graph of An against n and use your graph to estimate the time taken for the infested area to reach 10 000 ha.

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b Find the answer to a using logarithms. c Check your answer to b using suitable technology.

FINANCIAL GROWTH Suppose an amount u1 is invested at a rate of r% each compounding period. In this case the value of the investment after n periods is given by un+1 = u1 £ rn . In order to find n, the period of the investment, we need to use logarithms.

Example 28 Iryna has E5000 to invest in an account that pays 5:2% p.a. interest compounded annually. How long will it take for her investment to reach E20 000? Now un+1 = u1 £ rn ) 20 000 = 5000 £ (1:052)n ) (1:052)n = 4 ) log(1:052)n = log 4 ) n £ log 1:052 = log 4 log 4 ) n= ¼ 27:3 years log 1:052

un+1 = 20 000 after n years u1 = 5000 r = 105:2% = 1:052

) it will take at least 28 years.

EXERCISE 4H.2 1 A house is expected to increase in value at an average rate of 7:5% p.a. If the house is worth $160 000 now, how long is it expected to take for the value to reach $250 000? 2 Thabo has $10 000 to invest in an account that pays 4:8% p.a. compounded annually. How long will it take for his investment to grow to $15 000? 3 Dien invests $15 000 at 8:4% p.a. compounded monthly. He will withdraw his money when it reaches $25 000, at which time he plans to travel. The formula un+1 = u1 £ rn can be used to calculate the time needed where n is the time in months. a Explain why r = 1:007 . b After how many months can he withdraw the money? 4 Revisit the Opening Problem on page 102 and answer the questions posed.

DECAY EXERCISE 4H.3 1 The mass Mt of radioactive substance remaining after t years is given by Mt = 1000 £ e¡0:04t grams. Find the time taken for the mass to: a halve b reach 25 grams c reach 1% of its original value.

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2 A man jumps from an aeroplane and his speed of descent is given by V = 50(1¡e¡0:2t ) m s¡1 where t is the time in seconds. Find the time taken for his speed to reach 40 m s¡1 .

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3 The temperature T of a liquid which has been placed in a refrigerator is given by T = 4 + 96 £ e¡0:03t o C, where t is the time in minutes. Find the time required for the temperature to reach: a 25o C b 5o C. 4 The weight Wt of radioactive substance remaining after t years is given by Wt = 1000 £ 2¡0:04t grams. Find the time taken for the weight to: a halve b reach 20 grams c reach 1% of its original value. 5 The weight W (t) of radioactive uranium remaining after t years is given by the formula W (t) = W0 £ 2¡0:0002t grams, t > 0. Find the time taken for the original weight to fall to: a 25% of its original value b 0:1% of its original value. 6 The current I flowing in a transistor radio t seconds after it is switched off is given by I = I0 £ 2¡0:02t amps. Find the time taken for the current to drop to 10% of its original value. 7 A parachutist jumps from the basket of a stationary hot air balloon. His speed of descent is given by V = 50(1 ¡ 2¡0:2t ) m s¡1 where t is the time in seconds. Find the time taken for his speed to reach 40 m s¡1 .

REVIEW SET 4A 1 Find the following without using a calculator. Show all working. a

log4 64

b

log2 256

c

log2 (0:25)

d

log25 5

e

f

log6 6

g

log81 3

h

log9 (0:1)

i

log27 3

j

p 2 Without using a calculator, find: a log 10 3 Find x if:

log2 x = ¡3

a

4 Write as logarithmic equations:

1 b log p 3 10

b

log5 x ¼ 2:743

a

P = 3 £ bx

c b

log2 k ¼ 1:699 + x log A ¼ 5 log B ¡ 2:602

b

log 10a £ 10b+1

log3 x ¼ ¡3:145

m=

5 Write the following equations without logarithms: a c

c

n3 p2

loga Q = 3 loga P + loga R

6 Solve for x, giving your answer correct to 4 significant figures: b 20 £ 22x+1 = 500 a 5x = 7 7 The a b c d

log8 1 p logk k

¡

t

weight of radioactive substance after t years is Wt = 2500 £ 3 3000 grams. Find the initial weight. Find the time taken for the weight to reduce to 30% of its original value. Find the percentage weight loss after 1500 years. Sketch the graph of Wt against t.

8 Solve for x: 16x ¡ 5 £ 8x = 0 9 Solve the equation log3 (10x2 ¡ x ¡ 2) = 2 + 2 log3 x

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10 Find the exact value of a which satisfies the equation 53a £ 42a+1 = 103a+2 ln x Give your answer in the form where x, y 2 Z : ln y

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REVIEW SET 4B 1 Without using a calculator, find the base 10 logarithms of:

p 1000

a

10 p 3 10

b

10a 10¡b

c

2 Solve for x: a log x = 3

b

log3 (x + 2) = 1:732

c

log2 (

3 Write as a single logarithm: a log 16 + 2 log 3

b

log2 16 ¡ 2 log2 3

c

2 + log4 5

x ) = ¡0:671 10

4 Write the following equations without logarithms: a log T = 2 log x ¡ log y b log2 K = log2 n + 3x = 300

a

5 Solve for x:

b

30 £ 51¡x = 0:15

c

1 2

log2 t

3x+2 = 21¡x

6 If A = log2 2 and B = log2 3, write the following in terms of A and B: p b log2 54 c log2 (8 3) d log2 (20:25) e log2 (0:¹8) a log2 36 7 For a b c d e

the function g : x 7! log3 (x + 2) ¡ 2 : Find the domain and range. Find any asymptotes and axes intercepts for the graph of the function. Sketch the graph of y = g(x): Find g ¡1 . Explain how to verify your answer for g¡1 . Sketch the graphs of g, g ¡1 and y = x on the same axes. 3

8 Solve exactly for a in the equation log4 a5 + log2 a 2 = log8 625: 9 A straight line has equation y = mx + c. Its gradient is ¡2 and it passes through ¢ ¡ 3 . the point 1, log5 25 a Find the equation of the line. b If y = log5 M , find an expression for M in terms of x. c Hence, find the value of x when M = 25. y

4x £ 2y = 16 and 8x = 2 2 .

10 Solve simultaneously for x and y: p 11 Solve log8 4 x2 + 7 = 13

REVIEW SET 4C 1

a On the same set of axes sketch and clearly label graphs of: g : x 7! e¡x and h : x 7! ¡e¡x . f : x 7! ex , b What is the geometric connection between: i f and g

2 Sketch on the same set of axes the graphs of y = ex

ii

g and h?

and y = 3ex .

3 A particle moves in a straight line such that its displacement from the origin O is ¡t given by s(t) = 120t ¡ 40e 5 metres, where t is the time in seconds, t > 0. a Find the position of the particle at i

t = 0 ii ¡t 5

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b Hence sketch the graph of s(t) = 120t ¡ 40e

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4 Without using a calculator, find: 5 Simplify:

ln(e2x )

a

ln(e5 )

a

b

ln(e2 ex )

b

6 Write as a single logarithm: a ln 6 + ln 4 b ln 60 ¡ ln 20

c

c

ln

p ln( e) ³e´

ln

µ ¶ 1 e

ex ln 200 ¡ ln 8 + ln 5

d

ln 4 + ln 1

c

7 Write in the form a ln k where a and k are positive whole numbers and k is prime: a ln 32 b ln 125 c ln 729 8 Solve for x, giving answers correct to 3 significant figures: ex = 400

a

9 Solve 12(2x ) = 7 +

x

e2x+1 = 11

b

25e 2 = 750

c

e2x = 7ex ¡ 12

d

10 giving your answer in the form m + log2 n, m, n 2 Z . 2x

REVIEW SET 4D 1 On the same set of axes, sketch and clearly label the graphs of: g : x 7! e¡x , h : x 7! e¡x ¡ 4. f : x 7! ex , State the domain and range of each function. 2 Sketch on the same set of axes, the graphs of y = ex and y = e3x . 3 Without using a calculator, find: µ ¶ p 1 b ln 3 a ln(e e) e 4 Write in the form ex : 5 Simplify: a 4 ln 2 + 2 ln 3

a

c b

20 1 2

b

µ ln

e p e5

¶

3000

c

c

2 ln 5 ¡ 1

ln 9 ¡ ln 2

0:075 d

1 4

ln 81

6 Write the following equations without logarithms: a ln P = 1:5 ln Q + ln T b ln M = 1:2 ¡ 0:5 ln N 7 Consider g : x 7! 2ex ¡ 5. a Find the defining equation of g ¡1 . b Sketch the graphs of g and g¡1 on the same set of axes. c State the domain and range of g and g ¡1 . 8 The weight Wt grams of radioactive substance remaining after t weeks is given by ¡ t 20

Wt = 8000 £ e a halve

grams. Find the time for the weight to: b reach 1000 g c reach 0:1% of its original value.

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9 The function f is defined for x > 4 by f (x) = ln(x2 ¡ 16) ¡ ln x ¡ ln(x ¡ 4) µ ¶ x+a a Express f(x) in the form ln , stating the value of a 2 Z : x b Find an expression for f ¡1 (x).

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Chapter

Graphing and transforming functions Contents:

A B C D

Families of functions Transformations of graphs Simple rational functions Further graphical transformations

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Review set 5A Review set 5B

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GRAPHING AND TRANSFORMING FUNCTIONS (Chapter 5)

A

FAMILIES OF FUNCTIONS

There are several families of functions that you are already familiar with. These include: Name

General form

Function notation

Linear

f (x) = ax + b, a 6= 0

f : x 7! ax + b, a 6= 0

Quadratic

f (x) = ax2 + bx + c, a 6= 0

f : x 7! ax2 + bx + c, a 6= 0

Cubic

f (x) = ax3 + bx2 + cx + d, a 6= 0 f : x 7! ax3 + bx2 + cx + d, a 6= 0

Absolute value f (x) = jxj

f : x 7! jxj

Exponential

f (x) = a , a > 0, a 6= 1

f : x 7! ax , a > 0, a 6= 1

Logarithmic

f (x) = loge x or f(x) = ln x

f : x 7! ln x

Reciprocal

f (x) =

x

k , x 6= 0 x

f : x 7!

k , x 6= 0 x

These families of functions have different and distinctive graphs. We can compare them by considering important graphical features such as: ² the axes intercepts (where the graph cuts the x and y-axes) ² slopes ² turning points (maxima and minima) ² values of x where the function does not exist ² the presence of asymptotes (lines or curves that the graph approaches).

INVESTIGATION

FUNCTION FAMILIES

In this investigation you are encouraged to use the graphing package supplied.¡ Click on the icon to access this package.

GRAPHING PACKAGE

What to do:

1 From the menu, graph on the same set of axes: y = 2x + 1, y = 2x + 3, y = 2x ¡ 1 Comment on all lines of the form y = 2x + b. 2 From the menu, graph on the same set of axes: y = x + 2, y = 2x + 2, y = 4x + 2, y = ¡x + 2, y = ¡ 12 x + 2 Comment on all lines of the form y = ax + 2. 3 On the same set of axes graph: y = x2 , y = 2x2 , y = 12 x2 , y = ¡x2 , y = ¡3x2 , y = ¡ 15 x2 Comment on all functions of the form y = ax2 , a 6= 0.

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4 On the same set of axes graph: y = x2 , y = (x ¡ 1)2 + 2, y = (x + 1)2 ¡ 3, y = (x ¡ 2)2 ¡ 1 and other functions of the form y = (x ¡ h)2 + k of your choice. Comment on the functions of this form.

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5 On a b c

the same set of axes, graph these absolute value functions: y = j x j, y = 2 j x j, y = j 2x j y = j x j, y = j x j + 2, y = j x j ¡ 3 y = j x j, y = j x ¡ 2 j, y = j x + 3 j, y = j x ¡ 1 j + 2

Write a brief report on your discoveries. 6 On the same set of axes, graph these functions: 3 10 ¡1 ¡2 ¡5 1 b y= , y= , y= a y= , y= , y= x x x x x x 1 1 1 1 1 1 c y= , y= , y= d y = , y = + 2, y = ¡ 2 x x¡2 x+3 x x x e

y=

2 2 2 , y= + 2, y = ¡1 x x¡1 x+2

Write a brief report on your discoveries.

Example 1 If f (x) = x2 , find in simplest form: ³x´ a f(2x) b f c 2 f (x) + 1 3 ³x´ b f c 2 f (x) + 1 a f(2x) 3 2 = 2x2 + 1 ³ x ´2 = (2x) = = 4x2 3 =

d

f(x + 3) ¡ 4

d

f(x + 3) ¡ 4 = (x + 3)2 ¡ 4 = x2 + 6x + 9 ¡ 4 = x2 + 6x + 5

x2 9

EXERCISE 5A 1 If f(x) = x, find in simplest form: a

b

f(2x)

f (x) + 2

c

1 2

f (x)

d

2 f(x) + 3

c

f (x + 1)

d

2 f(x + 1) ¡ 3

2 If f(x) = x3 , find in simplest form: a

1 2

b

f(4x)

f(2x)

Note: (x + 1)3 = x3 + 3x2 + 3x + 1. See the binomial theorem, Chapter 8. 3 If f(x) = 2x , find in simplest form: a

f (¡x) + 1

c

f (x ¡ 2) + 3

d

2 f(x) + 3

1 , find in simplest form: x f(¡x) b f ( 12 x)

c

2 f(x) + 3

d

3 f(x ¡ 1) + 2

b

f(2x)

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4 If f(x) =

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For the following questions, use the graphing package or your graphics calculator to graph and find the key features of the functions.

GRAPHING PACKAGE

5 Consider f : x 7! 2x + 3 or y = 2x + 3. a Graph the function. b Find algebraically, the: i x-axis intercept ii y-axis intercept c Use technology to check the axes intercepts found in b.

iii

slope.

6 Consider f : x 7! (x ¡ 2)2 ¡ 9. a Graph the function. b Find algebraically the x and y axes intercepts. c Use technology to check that: i the x-axis intercepts are ¡1 and 5 ii iii the vertex is (2, ¡9).

the y-intercept is ¡5

7 Consider f : x 7! 2x3 ¡ 9x2 + 12x ¡ 5. a Graph the function. b Check that: i the x-intercepts are 1 and 2 12 ii the y-intercept is ¡5 iii the minimum turning point is at (2, ¡1) iv the maximum turning point is at (1, 0). 8 Sketch the graph of y = j x j. Note: j x j = x if x > 0 and j x j = ¡x if x < 0. 9 Consider f : x 7! 2x . Graph the function and check these key features: a as x ! 1, 2x ! 1 b as x ! ¡1, 2x ! 0 (from above) c the y-intercept is 1 d 2x is > 0 for all x.

! reads ‘approaches’ or ‘tends to’

10 Consider f : x 7! loge x. Graph the function and then check that: a as x ! 1, ln x ! 1 b as x ! 0 (from the right), ln x ! ¡1 c ln x only exists if x > 0 d the x-intercept is 1 e the y-axis is an asymptote.

B

TRANSFORMATIONS OF GRAPHS

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In the next exercise you should discover the graphical connection between y = f(x) and functions of the form:

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TYPES y = f(x) + b AND y = f(x ¡ a) EXERCISE 5B.1 a Sketch the graph of f(x) = x2 . b On the same set of axes sketch the graphs of: i y = f(x) + 2, i.e., y = x2 + 2 ii y = f (x) ¡ 3, i.e., y = x2 ¡ 3

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c What is the connection between the graphs of y = f(x) and y = f (x) + b if: i b>0 ii b < 0? 2 For each of the following functions f, sketch on the same set of axes y = f (x), y = f (x) + 1 and y = f(x) ¡ 2. 1 c f (x) = x3 d f (x) = a f(x) = j x j b f (x) = 2x x Summarise your observations by describing the graphical transformation of y = f(x) as it becomes y = f (x) + b. a On the same set of axes, graph: f(x) = x2 , y = f(x ¡ 3) and y = f(x + 2).

3

b What is the connection between the graphs of y = f(x) and y = f (x ¡ a) if: i a>0 ii a < 0? 4 For each of the following functions f, sketch on the same set of axes the graphs of y = f (x), y = f(x ¡ 1) and y = f (x + 2). 1 c f (x) = ln x d f (x) = a f(x) = j x j b f (x) = x3 x Summarise your observations by describing the geometrical transformation of y = f(x) as it becomes y = f (x ¡ a). 5 For each of the following functions sketch: y = f (x), y = f(x ¡ 2) + 3 and y = f (x + 1) ¡ 4 on the same set of axes. 1 a f(x) = x2 b f(x) = ex c f(x) = x 6 Copy these functions and then draw the graph of y = f(x ¡ 2) ¡ 3: a b y

y

x x

7 Given f(x) = x2

is transformed to g(x) = (x ¡ 3)2 + 2 :

a find the images of the following points on f(x) : i (0, 0) ii where x = ¡3

iii

where x = 2

b find the points on f (x) which correspond to the following points on g(x) :

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TYPES y = p f(x) , p > 0

AND y = f(kx) , k > 0

EXERCISE 5B.2 1 Sketch on the same set of axes, the graphs of y = f (x), y = 2 f (x) and y = 3 f (x) for each of: a

f(x) = x2

b

f(x) = j x j

c

d

f(x) = ex

e

f(x) = ln x

f

f(x) = x3 1 f(x) = x

2 Sketch on the same set of axes, the graphs of y = f (x), y = 12 f(x) and y = 14 f (x) for each of: a

f(x) = x2

f(x) = x3

b

c

f(x) = ex

3 Using 1 and 2, summarise your observations by describing the graphical transformation of y = f (x) to y = p f(x) for p > 0. 4 Sketch on the same set of axes, the graphs of y = f (x) and y = f(2x) for each of: b y = (x ¡ 1)2 c y = (x + 3)2 a y = x2 5 Sketch on the same set of axes, the graphs of y = f (x) and y = f(3x) for each of: c y = ex a y=x b y = x2 ¡ ¢ 6 Sketch on the same set of axes, the graphs of y = f (x) and y = f x2 for each of: a

y = x2

b

c

y = 2x

y = (x + 2)2

7 Using 4, 5 and 6, summarise your observations by describing the graphical transformation of y = f (x) to y = f (kx) for k > 0. 8 Consider the function f : x 7! x2 . On the same set of axes sketch the graphs of: a y = f(x), y = 3 f(x ¡ 2) + 1 and y = 2 f(x + 1) ¡ 3 ¢ ¡ ¢ ¡ ¢ ¡ b y = f(x), y = f(x ¡ 3), y = f x2 ¡ 3 , y = 2 f x2 ¡ 3 and y = 2 f x2 ¡ 3 + 4 c y = f(x) and y = 9

1 4

f(2x + 5) + 1.

a Given that the following points lie on y = f (x), find the corresponding points on the image function y = 3f (2x) : i (3, ¡5) ii (1, 2) iii (¡2, 1) b Find the points on y = f (x) which are moved to the following points under the transformation y = 3f (2x) : i (2, 1) ii (¡3, 2) iii (¡7, 3)

10 The function y = f(x) is transformed to the function g(x) = 3 + 2f ( 12 x + 1):

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a Fully describe the transformation that maps f(x) onto g(x). b Using a, find the image points of the following points on f(x): i (1, ¡3) ii (2, 1) iii (¡1, ¡2) c Find the points on f (x) which correspond to the following points on g(x): i (¡2, ¡5) ii (1, ¡1) iii (5, 0)

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TYPES y = ¡f(x) AND y = f(¡x) EXERCISE 5B.3 1 On the same set of axes, sketch the graphs a y = 3x and y = ¡3x c y = x2 and y = ¡x2 e y = x3 ¡ 2 and y = ¡x3 + 2

of: b y = ex and y = ¡ex d y = ln x and y = ¡ ln x f y = 2(x + 1)2 and y = ¡2(x + 1)2

2 Based on question 1, what transformation moves y = f (x) to y = ¡f (x)? a Find f (¡x) for:

3

i

ii

f (x) = 2x + 1

f(x) = x2 + 2x + 1

iii

f (x) = j x ¡ 3 j

iii

f (x) = j x ¡ 3 j

b Graph y = f (x) and y = f (¡x) for: i

ii

f (x) = 2x + 1

f(x) = x2 + 2x + 1

4 Based on question 3, what transformation moves y = f (x) to y = f (¡x)? 5 The function y = f (x) is transformed to g(x) = ¡f (x): a Find the points on g(x) corresponding to the following points on f (x): i (3, 0) ii (2, ¡1) iii (¡3, 2) b Find the points on f (x) that have been transformed to the following points on g(x): i (7, ¡1) ii (¡5, 0) iii (¡3, ¡2) 6 The function y = f (x) is transformed to h(x) = f(¡x): a Find the image points on i (2, ¡1) ii b Find the points on f (x) i (5, ¡4) ii

h(x) for the following points on f(x): (0, 3) iii (¡1, 2) corresponding to the following points on h(x): (0, 3) iii (2, 3)

7 A function y = f(x) is transformed to the function y = ¡f (¡x) = g(x): a Describe the nature of the transformation. b If (3, ¡7) lies on y = f(x), find the transformed point on g(x). c Find the point on f (x) that transforms to the point (¡5, ¡1).

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For y = f (x ¡ a) + b, the graph is translated horizontally a units and £ ¤ vertically b units. We say it is translated by the vector ab : continued next page

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For y = f (x ¡ a), the effect of a is to translate the graph horizontally through a units. ² If a > 0 it moves to the right. ² If a < 0 it moves to the left.

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For y = f (x) + b, the effect of b is to translate the graph vertically through b units. ² If b > 0 it moves upwards. ² If b < 0 it moves downwards.

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Summary of graphical transformations on y = f (x)

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continued from previous page

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For y = p f (x), p > 0, the effect of p is to vertically stretch the graph by a factor of p. ² If p > 1 it moves points of y = f (x) further away from the x-axis. ² If 0 < p < 1 it moves points of y = f (x) closer to the x-axis.

I

For y = f (kx), k > 0, the effect of k is to horizontally compress the graph by a factor of k. ² If k > 1 it moves points of y = f(x) closer to the y-axis. ² If 0 < k < 1 it moves points of y = f (x) further away from the y-axis.

I I

For y = ¡f (x), the effect is to reflect y = f (x) in the x-axis. For y = f(¡x), the effect is to reflect y = f (x) in the y-axis.

Note:

Stretching by a factor of p1 is equivalent to compressing by a factor of p. For example, y = 13 f (x) indicates that y = f(x) is vertically stretched by a factor of 13 , or compressed by a factor of 3. Likewise, y = f ( 14 x) indicates that y = f (x) is horizontally compressed by a factor of 14 , or stretched by a factor of 4.

EXERCISE 5B.4 1 Copy the following graphs for y = f(x) and sketch the graphs of y = ¡f(x) on the same axes. a b c y y y x

x 2

1 x

2 Given the following graphs of y = f (x), sketch graphs of y = f (¡x) : a b c y y y y=1 1 x

x x x=2

3 The scales on the graphs below are the same. Match each equation to its graph. y = x4

y = 2x4

B b

y

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y = 12 x4

C c

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4 For the graph of y = f(x) given, draw sketches of: a y = 2f (x) b y = 12 f (x) c y = f (x + 2) d y = f (2x) 1 e y = f ( 2 x)

y

5

y

1

3

c 2

x

3

For the graph of y = g(x) given, draw sketches of: a y = g(x) + 2 b y = ¡g(x)

g(x)

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d

y = g(¡x)

y = g(x + 1)

x

y

6 For the graph of y = h(x) given, draw sketches of: a y = h(x) + 1 b y = 12 h(x) ¡ ¢ c y = h(¡x) d y = h x2

y=h(x)

3

-1 -2

C

x

(2,-2)

SIMPLE RATIONAL FUNCTIONS ax + b d , x 6= ¡ cx + d c is called a simple rational function. Any function x 7!

where a, b, c and d are constants

These functions are characterised by the presence of both a horizontal asymptote (HA) and a vertical asymptote (VA). Any graph of a simple rational function can be 1 obtained from the reciprocal function x 7! x by a combination of transformations including: ²

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Example 2 a

Find the function y = g(x) that results when transforming the reciprocal 1 function, x 7! by: a vertical stretch with factor 2 then a horizontal x h i 3 : compression with factor 3, then a translation of ¡2

b c

Find the asymptotes of each function found in a. Is the function found in a a self inverse function? Explain.

a

Under a vertical stretch with factor 2, Under a horizontal compression with factor 3, h i 3 , Under a translation of ¡2 )

b

f (x) =

1 x

µ ¶ 1 becomes 2 x

2 x

becomes

2 3x

2 (3x) 2 ¡2 3(x ¡ 3)

becomes

f2f(x)g ff (3x)g ff (x ¡ 3) ¡ 2g

2 2(3x ¡ 9) ¡6x + 20 2 ¡2= ¡ = 3x ¡ 9 3x ¡ 9 3x ¡ 9 3x ¡ 9

1 are: VA x = 0, HA y = 0 x h i 3 g ) for the new function VA is x = 3, HA is y = ¡2 fas translated ¡2 The asymptotes of y =

c

y

From a graphics calculator the graph is found as shown.

y=x

It is not symmetrical about y = x. Hence, it is not a self inverse function. x

y=-2 x=3

Note: f(x) =

k x

is a vertical stretch of x 7!

1 x

with factor k.

EXERCISE 5C

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ax + b 1 a Find, in the form y = , the function that results when x 7! cx + d x is transformed by: i a vertical stretch of factor 12 ii a horizontal stretch of factor 3 iii a horizontal translation of ¡3 iv a vertical translation of 4 v all of i, ii, iii and iv. ax + b b Find the domain and range of y = as found in a v. cx + d

1

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Example 3 2x ¡ 6 , find: x+1 1 b how to transform the function to give x 7! x

For the function f : x 7! a the asymptotes

a

2x ¡ 6 2(x + 1) ¡ 8 8 ¡8 = =2¡ = +2 x+1 x+1 x+1 x+1 £ ¤ ¡8 from f (x) = This represents a translation of ¡1 which has 2 x VA x = 0 and HA y = 0. f (x) =

So, f (x) = Note:

2x ¡ 6 x+1

has VA x = ¡1 and HA y = 2.

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2x ¡ 6 is undefined when x = ¡1 x+1

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as jxj ! 1, f (x) ! 2

²

the domain of f (x) = the range of f(x) =

b

To get f(x) =

2x ¡ 6 x+1

2x ¡ 6 x+1

2x ¡ 6 x+1

is fy: y 6= 2g

from f (x) =

1 x

we

µ ¶ 1 8 1 becomes 8 = g f x x x 8 8 f becomes ¡ g x x ¡8 ¡8 f becomes + 2g x x+1

vertically stretch by a factor of 8 then reflect in the x-axis £ ¡1 ¤

then translate by

is fx: x 6= ¡1g

2

So, to do the opposite we h i 1 , then reflect in the x-axis, then vertically stretch by factor 18 . translate by ¡2 2x ¡ 6 2(x ¡ 1) ¡ 6 2x ¡ 8 8 becomes y = ¡2= ¡2 =¡ x+1 (x ¡ 1) + 1 x x 8 8 8 8 1 then ¡ becomes and becomes = . x x x (8x) x

Check: y =

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b f : x 7!

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how to transform f(x) to give the function x 7!

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2 For these functions find: i the asymptotes ii

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Example 4 a Find the asymptotes of y = f (x): 4x + 3 . b Find the axes intercepts. x¡2 Discuss the behaviour of f near its i VA ii HA. Sketch the graph of the function. 1 4x + 3 Describe the transformations which move x 7! to x 7! . x x¡2

Consider f(x) = c d e a

y=

4(x ¡ 2) + 11 11 4x + 3 = =4+ x¡2 x¡2 x¡2 fwhere y is undefinedg fas jxj ! 1, y ! 4g

So, the function has VA x = 2 and has HA y = 4 b

When x = 0, y =

3 ¡2

) y-intercept is ¡1 12

when y = 0, 4x + 3 = 0 x = ¡ 34 c

) x-intercept is ¡ 34

i as x ! 2 (from the left), y ! ¡1 ii as x ! ¡1, y ! 4 (from below) as x ! 2 (from the right), y ! 1 as x ! 1, y ! 4 (from above) y

d

e

y=4

x -\Er_

1 11 becomes under a vertical x x stretch with factor 11, and then 11 4x + 3 becomes under a x x¡2 £ ¤ translation of 24

x=2

-1\Qw_

3 For the following functions: i find the asymptotes ii find the axes intercepts iii discuss the graph’s behaviour near its VA and its HA iv sketch the graph. 1 v Describe the transformations which move x 7! to the given function. x 2x + 3 3 2x ¡ 1 5x ¡ 1 a y= b y= c y= d y= x+1 x¡2 3¡x 2x + 1

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4 In order to remove noxious weeds from her property Helga sprays with a weedicide. The chemical is slow acting and the number of weeds per hectare remaining after t days is 100 modelled by N = 20 + weeds/ha. t+2 a How many weeds per ha were alive before the spraying? b How many weeds will be alive after 8 days? c How long will it take for the number of weeds still alive to be 40/ha? d Sketch the graph of N against t e According to the model, is the spraying going to eradicate all weeds?

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Graphics calculator tips: ²

To find the zeros of a function y = f (x), simply graph the function and find its x-intercepts. This is equivalent to finding the roots or solutions of the equation f(x) = 0:

²

To check that you have found the correct asymptotes (VA and HA): I Try to find y for the x-asymptote value. It should be undefined. I Try to find y for large x values, e.g., §109 . It should give a value close to the y-asymptote value.

D FURTHER GRAPHICAL TRANSFORMATIONS In this exercise you should discover the graphical connection between y = f(x) and 1 functions of the form y = , y = j f(x) j and y = f (j x j): f (x)

THE RECIPROCAL FUNCTION y =

1 f (x)

Example 5 Graph on the same set of axes: 1 a y = x ¡ 2 and y = x¡2 a

y

b

y = x2

b

y = x-2

and y =

1 x2 y y = x2

2 y=

x 1 x-2

y=

-\Qw_ -2

1 x2

x

x=2

EXERCISE 5D.1 1 Graph on the same set of axes: ¡1 a y = ¡x2 and y = 2 x

y = (x ¡ 1)(x ¡ 3) and y =

b

1 (x ¡ 1)(x ¡ 3)

2 Invariant points are points which do not move under a transformation. 1 Show that if y = f(x) is transformed to y = , invariant points occur at y = §1. f(x)

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Check your results in question 1 for invariant points.

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DISCUSSION

THE GRAPHICAL CONNECTION BETWEEN 1 y = f (x) AND y = f (x) True or false? Discuss:

²

²

² ²

1 The zeros of f (x) become VA values of and the VA values of f(x) f(x) 1 . become the zeros of f (x) 1 and the minimum Maximum values of f(x) become minimum values of f(x) 1 . values of f (x) become maximum values of f(x) 1 1 > 0 also and when f (x) < 0, < 0 also. When f (x) > 0, f (x) f(x) 1 1 ! §1 and when ! 0, f(x) ! §1: When f (x) ! 0, f (x) f(x)

THE MODULUS FUNCTIONS y = jf (x)j

AND y = f (jxj)

Example 6 Draw the graph of f(x) = 3x(x ¡ 2) and on the same axes draw the graphs of: a y = jf(x)j b y = f (jxj) ( a

y = jf (x)j =

(

f(x) if f (x) > 0 ¡f (x) if f (x) < 0

b y = f (jxj) =

This means the graph is unchanged for f(x)¡>¡0, reflected in the x-axis for f(x)¡ 0 f(¡x) if x < 0

This means the graph is unchanged if x¡>¡0, reflected in the y-axis if x¡ 0 and y = ¡f (x) for f(x) < 0

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True or false? Discuss:

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²

Points on the y-axis are invariant for f(x) ! j f(x) j

²

The graph of y = f (j x j) can be obtained from the graph of y = f (x) by reflecting in the y-axis.

²

y = f (j x j) )

²

Points on the x-axis are invariant for f (x) ! f (j x j)

y = f (x) for x > 0 and y = f (¡x) for x < 0:

EXERCISE 5D.2 1 Draw y = x(x + 2) and on the same set of axes graph: a y = j f (x) j b y = f (j x j) 1 2 Copy the following graphs for y = f(x) and on the same axes graph y = : f (x) a b c y

y

y

y¡=¡¦(x)

y¡=¡¦(x)

y¡=¡¦(x) y=1

1

x

3

x

x

x=4

(1\Qw_\'-2)

3 Copy the following graphs for y = f(x) and on the same axes graph y = j f (x) j : a b c y y

y

y¡=¡¦(x)

y¡=¡¦(x)

y¡=¡¦(x)

-2

2

x

y=2

2

x

x

-2 x=4

4 Repeat question 3, but this time graph y = f (j x j) instead of y = j f (x) j . 5 Suppose the function f (x) is transformed to j f(x) j : For the following points on f (x) find the image points on j f (x) j : a (3, 0) b (5, ¡2) c (0, 7) d

(2, 2)

6 Suppose the function f (x) is transformed to f (j x j) : a For the following points on f (x) find the image points: i (0, 3) ii (1, 3) iii

(7, ¡4)

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b For the points on f (j x j) find the points on f(x) that have been transformed: i (0, 3) ii (¡1, 3) iii (10, ¡8)

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REVIEW SET 5A 1 If f (x) = x2 ¡ 2x, find in simplest form: a f (3) b f(¡2) c f (2x)

d

f (¡x)

2 If f (x) = 5 ¡ x ¡ x2 , find in simplest form:

3f(x) ¡ 2

e

³x´

e 2f(x) ¡ f(¡x) 2 3 Consider the function f : x 7! x2 . On the same set of axes graph: a y = f(x) b y = f (x + 2) c y = 2f(x + 2) d y = 2f(x + 2) ¡ 3 a

b

f (4)

c

f(¡1)

f (x ¡ 1)

d

f

4 The graph of f(x) = 3x3 ¡ 2x2 + x + 2 is translated to its image g(x) by the £1¤ . Write the equation of g(x) in the form g(x) = ax3 + bx2 + cx + d. vector ¡2 5 Consider f(x) = (x + 1)2 ¡ 4. a b c d

Use your calculator to help graph the function. Find algebraically i the x-intercepts ii the y-intercept. What are the coordinates of the vertex of the function? Use your calculator to check your answers to b and c.

6 Consider f : x 7! 2¡x . a Use your calculator to help graph the function. b True or false: i as x ! 1, 2¡x ! 0 ii iv iii the y-intercept is 12

as x ! ¡1, 2¡x ! 0 2¡x > 0 for all x?

7 Sketch the graph of f (x) = ¡x2 , and on the same set of axes sketch the graph of: a y = f(¡x) b y = ¡f(x) c y = f (2x) d y = f (x ¡ 2) 8 The graph of y1 = f (x) is shown alongside. The x-axis is a tangent to f (x) at x = a and f(x) cuts the x-axis at x = b.

y

On the same diagram sketch the graph of y2 = f (x ¡ c) where 0 < c < b ¡ a. Indicate the coordinates of the points of intersection of y2 with the x-axis.

a

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9 The graph of f (x) is drawn alongside. a Copy the graph of f (x) and draw ¦(x) the graph of j f (x) j on the same 2 set of axes. 1 1 . b Find the y-intercept of f (x) 1 2 c Show on the diagram the points that -1 (2,-1) 1 are invariant for the function : f (x) 1 on the same set of axes. d Draw the graph of y = f (x)

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10

a If f (x) = x + 2, find the equation of the function F obtained by stretching the function f vertically by a factor of 2 and compressing the function horizontally by a factor of 2, followed by a translation of 12 horizontally and ¡3 vertically. i Check that the point (1, 3) remains invariant under the transformation described in part a.

b

ii What happens to the points (0, 2) and (¡1, 1) under the transformation given in part a? iii Show that the points in ii lie on the graph of y = F (x): y

11 The graph of f(x) = x2 is transformed to the graph of g(x) by a reflection and a translation as illustrated in the diagram alongside. Find the equation of y = g(x) in the form ax2 + bx + c.

y=¦(x)

V(-3,¡2)

x y=g(x)

REVIEW SET 5B 4 , find in simplest form: x ³x´ f (¡4) b f(2x) c f 2

1 If f (x) = a

4f (x + 2) ¡ 3

d

2 Consider f(x) : x 7! 3x ¡ 2. a Sketch the function f: b Find algebraically the i x-intercept ii y-intercept iii slope. c i Find y when x = 0:3 ii Find x when y = 0:7 3 For what values of x, where a is a positive real number, is j x ¡ a j = j x j ¡ a ? c , x 6= ¡c, c > 0. x+c a On a set of axes like those shown, sketch the graph of f(x). Label clearly any points of intersection with the axes and any asymptotes. 1 b On the same set of axes, sketch the graph of . f (x)

4 Let f(x) =

Label clearly any points of intersection with the axes. 5 For the graph of y = f(x), sketch graphs of: a y = f(¡x) b y = ¡f(x) c y = f(x + 2) d y = f(x) + 2

(-3, 3)

y

-c

c

y

3

x

(5, 8)

y¡=¡¦(x) x

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6 The graph of y = f(x) is given. On the same set of axes graph each pair of functions: a y = f (x) and y = f (x ¡ 2) + 1 1 b y = f (x) and y = f (x)

y

y¡=¡¦(x) (2,-2)

c y = f (x) and y = j f (x) j

x

x¡=¡4

a Find the equation of the function y = f(x) that results from transforming the 1 function x 7! by: a reflection in the y-axis, then a vertical stretch of x £ ¤ factor 3, then a horizontal compression of factor 2, then a translation of 11 : b Sketch y = f (x) and state its domain and range. c Does y = f(x) have an inverse function? Explain. d Is the function f a self-inverse function? Give graphical and algebraic evidence.

7

2x ¡ 3 . 3x + 5 Your discussion should include: the asymptote equations, axes intercepts, and what happens near the asymptotes.

8 Discuss and sketch the graph of y =

y

9 The graph alongside is that of a rational relation. ax + b : Find its equation in the form y = cx + d

x=3

x y = -2

10 For each of the following functions: h i 1 i find y = f (x), the result of a translation ¡2 : ii Sketch the original function and its translated function on the same set of axes. iii Clearly state any asymptotes of each function. iv State the domain and range of each function. b y = log4 x y = 2x

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a Sketch the graph of f (x) = ¡2x + 3, clearly showing the axis intercepts. 1 b Find the invariant points for the graph of y = : f (x) c State the equation of the vertical asymptote and find the y-intercept of the graph 1 of y = . f(x) 1 on the same axes as in part a, showing clearly d Sketch the graph of y = f (x) the information found in parts b and c. e On a new pair of axes, sketch the graphs of y = j f (x) j and y = f (j x j) showing clearly all important features.

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Chapter

6

Quadratic equations and functions A B C D E F G H

Contents:

Solving quadratic equations (Review) The discriminant of a quadratic The sum and product of the roots Graphing quadratic functions Finding a quadratic from its graph Where functions meet Problem solving with quadratics Quadratic optimisation

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Review set 6A Review set 6C Review set 6E

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Review set 6B Review set 6D

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QUADRATIC EQUATIONS AND FUNCTIONS (Chapter 6)

QUADRATICS A quadratic equation is an equation of the form ax2 + bx + c = 0 where a, b and c are constants, a 6= 0. A quadratic function is a function of the form f (x) = ax2 + bx + c, a 6= 0. Alternatively, it can be written as f : x 7! ax2 + bx + c, a 6= 0. Quadratic functions are members of the family of polynomials. f : x 7! ax3 + bx2 + cx + d, a 6= 0 is a cubic polynomial. f : x 7! ax4 + bx3 + cx2 + dx + e, a 6= 0 is a quartic polynomial.

HISTORICAL NOTE Galileo Galilei (1564 - 1642) was born in Pisa, Tuscany. He was a philosopher who played a significant role in the scientific revolution of that time. Within his research he conducted a series of experiments on the paths of projectiles, attempting to find a mathematical description of falling bodies. Two of Galileo’s experiments consisted of rolling a ball down a grooved ramp that was placed at a fixed height above the floor and inclined at a fixed angle to the horizontal. In one experiment the ball left the end of the ramp and descended to the floor. In the second, a horizontal shelf was placed at the end of the ramp, and the ball travelled along this shelf before descending to the floor. In each experiment Galileo altered the release height (h) of the ball and measured the distance (d) the ball travelled before landing. The units of measurement were called ‘punti’ (points). In both experiments Galileo found that once the ball left the ramp or shelf, its path was parabolic and could therefore be modelled by a quadratic function.

Galileo

OPENING PROBLEM Farmer Brown wishes to construct a rectangular pen for her chickens using her barn for one of the sides. The other three sides will be constructed out of 80 m of chicken wire as illustrated. She can make the pen long and thin, or short and fat, or any rectangular shape in between, as long as she uses exactly 80 m of chicken wire. However, what she really wants to know is which rectangular shape will give her chickens the maximum area.

pen

barn

Can you: ² decide on a suitable variable to use and construct an area function in terms of this variable

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² use the area function to find the maximum area that the pen may have and the shape of the pen when this occurs?

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A

SOLVING QUADRATIC EQUATIONS (REVIEW)

Acme Leather Jacket Co. makes and sells x leather jackets each week and their profit function is given by P = ¡12:5x2 + 550x ¡ 2125 dollars. How many jackets must be made and sold each week in order to obtain a weekly profit of $3000? Clearly we need to solve the equation: ¡12:5x2 + 550x ¡ 2125 = 3000 i.e., 12:5x2 ¡ 550x + 5125 = 0 This equation is of the form ax2 + bx + c = 0 and is thus a quadratic equation. To solve quadratic equations we can: ² factorise the quadratic and use the Null Factor law: “if ab = 0 then a = 0 or b = 0” ² complete the square ² use the quadratic formula ² use technology. Definition: The roots or solutions of ax2 + bx + c = 0 are the values of x which satisfy the equation, i.e., make it true. For example, x = 2 is a root of x2 ¡ 3x + 2 = 0 since, when x = 2, x2 ¡ 3x + 2 = (2)2 ¡ 3(2) + 2 = 4 ¡ 6 + 2 = 0 X

SOLVING BY FACTORISATION Step Step Step Step

1: 2: 3: 4:

Make one side of the equation 0 by transferring all terms to one side. Fully factorise the other side. Use the ‘Null Factor law’: “if ab = 0 then a = 0 or b = 0”. Solve the resulting elementary linear equations.

Example 1 a 3x2 + 5x = 0

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3x2 + 5x = 0 x(3x + 5) = 0 ) x = 0 or 3x + 5 = 0 ) x = 0 or x = ¡ 53

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b x2 = 5x + 6 x2 = 5x + 6 ) x2 ¡ 5x ¡ 6 = 0 ) (x ¡ 6)(x + 1) = 0 ) x ¡ 6 = 0 or x + 1 = 0 ) x = 6 or ¡1

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Solve for x:

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Example 2 a 4x2 + 1 = 4x

Solve for x: a

b 6x2 = 11x + 10

4x2 + 1 = 4x ) 4x2 ¡ 4x + 1 = 0 ) (2x ¡ 1)2 = 0 )

1 2

x=

Example 3

b

6x2 = 11x + 10 ) 6x2 ¡ 11x ¡ 10 = 0 ) (2x ¡ 5)(3x + 2) = 0 ) x = 52 or ¡ 23

2 = ¡7 x 2 ) x(3x + ) = ¡7x x 3x +

Solve for x: 2 = ¡7 x

3x +

fmultiply both sides by xg

fexpand the bracketsg ) 3x2 + 2 = ¡7x 2 fmake the RHS 0g ) 3x + 7x + 2 = 0 ) (x + 2)(3x + 1) = 0 ffactorisingg ) x = ¡2 or ¡ 13

EXERCISE 6A.1 1 Solve the following by factorisation: b 6x2 + 2x = 0 a 4x2 + 7x = 0 2 e 3x2 = 8x d 2x ¡ 11x = 0 g x2 ¡ 5x + 6 = 0 h x2 = 2x + 8 2 k x2 + x = 12 j 9 + x = 6x

c f i l

3x2 ¡ 7x = 0 9x = 6x2 x2 + 21 = 10x x2 + 8x = 33

2 Solve the following by factorisation: b 2x2 ¡ 13x ¡ 7 = 0 a 9x2 ¡ 12x + 4 = 0 2 e 2x2 + 3 = 5x d 3x + 5x = 2 h 4x2 + 4x = 3 g 3x2 = 10x + 8 2 k 7x2 + 6x = 1 j 12x = 11x + 15

c f i l

3x2 = 16x + 12 3x2 = 4x + 4 4x2 = 11x + 3 15x2 + 2x = 56

x+

e

2x ¡

f

x+3 9 =¡ 1¡x x

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3 Solve for x:

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SOLVING BY ‘COMPLETING THE SQUARE’ As you would be aware by now, not all quadratics factorise easily. For example, x2 + 4x + 1 cannot be factorised by using a simple factorisation approach. This means that we need a different approach in order to solve x2 + 4x + 1 = 0. One way is to use the ‘completing the square’ technique. Equations of the form ax2 + bx + c = 0 can be converted to the form (x + p)2 = q from which the solutions are easy to obtain.

Notice that if X 2 = a, pthen X = § a.

Example 4 a (x + 2)2 = 7

Solve exactly for x: a

(x + 2)2 = 7 p ) x+2 = § 7 p ) x = ¡2 § 7

b (x ¡ 1)2 = ¡5

(x ¡ 1)2 = ¡5 has no real solutions since the perfect square (x ¡ 1)2 cannot be negative.

b

Example 5 x2 + 4x + 1 = 0

Solve for exact values of x: x2 + 4x + 1 = 0 ) x2 + 4x = ¡1 x2 + 4x + 22 = ¡1 + 22 ) (x + 2)2 = 3 p ) x+2 = § 3 p ) x = ¡2 § 3

)

fput the constant on the RHSg fcompleting the squareg ffactorisingg The squared number we add to both sides is ¶2 µ coefficient of x 2

EXERCISE 6A.2 1 Solve for exact values of x: (x + 5)2 = 2 (x ¡ 8)2 = 7 (x + 1)2 + 1 = 11

a d g

b e h

(x + 6)2 = ¡11 2(x + 3)2 = 10 (2x + 1)2 = 3

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2 Solve for exact values of x by completing the square: b x2 + 6x + 2 = 0 a x2 ¡ 4x + 1 = 0 e x2 + 6x + 7 = 0 d x2 = 4x + 3 2 h x2 + 10 = 8x g x + 6x = 2

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c f i

(x ¡ 4)2 = 8 3(x ¡ 2)2 = 18 (1 ¡ 3x)2 ¡ 7 = 0

c f i

x2 ¡ 14x + 46 = 0 x2 = 2x + 6 x2 + 6x = ¡11

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If the coefficient of x2 is not 1, we first divide throughout to make it 1. For example, 2x2 + 10x + 3 = 0 ¡3x2 + 12x + 5 = 0

becomes

x2 + 5x +

becomes

x2 ¡ 4x ¡

3 2 5 3

3 Solve for exact values of x by completing the square: b 2x2 ¡ 10x + 3 = 0 a 2x2 + 4x + 1 = 0 e 5x2 ¡ 15x + 2 = 0 d 3x2 = 6x + 4

=0 =0

c f

3x2 + 12x + 5 = 0 4x2 + 4x = 5

THE QUADRATIC FORMULA Many quadratic equations cannot be solved by factorising, and completing the square can be rather tedious. Consequently, the quadratic formula has been developed. This formula is: 2

If ax + bx + c = 0, then x = Proof:

If ax2 + bx + c = 0, b c then x2 + x + = 0 a a b c ) x2 + x =¡ a a µ ¶2 µ ¶2 b b b c =¡ + ) x2 + x + a 2a a 2a µ ¶2 b2 ¡ 4ac b = ) x+ 2a 4a2 r b2 ¡ 4ac b ) x+ =§ 2a 4a2 p ¡b § b2 ¡ 4ac ) x= 2a

¡b §

p b2 ¡ 4ac . 2a

fdividing each term by a, as a 6= 0g

fcompleting the square on LHSg

For example, consider the Acme Leather Jacket Co. equation from page 145. We need to solve: 12:5x2 ¡ 550x + 5125 = 0 Trying to factorise so in this case a = 12:5, b = ¡550, c = 5125 this equation or p using ‘completing 550 § (¡550)2 ¡ 4(12:5)(5125) the square’ would ) x = not be easy. 2(12:5) p 550 § 46 250 = 25 ¼ 30:60 or 13:40

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However, x needs to be a whole number, so x = 13 or 31 would produce a profit of around $3000 each week.

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Example 6 a x2 ¡ 2x ¡ 6 = 0

Solve for x: a

b 2x2 + 3x ¡ 4 = 0

x2 ¡ 2x ¡ 6 = 0 has a = 1, b = ¡2, c = ¡6 p ¡(¡2) § (¡2)2 ¡ 4(1)(¡6) ) x= 2(1) p 2 § 4 + 24 ) x= 2 p 2 § 28 ) x= 2 p 2§2 7 ) x= 2 p ) x = 1§ 7 p p Solutions are: 1 + 7 and 1 ¡ 7:

b 2x2 + 3x ¡ 6 = 0 has a = 2, b = 3, c = ¡6 p ¡3 § 32 ¡ 4(2)(¡6) ) x= 2(2) p ¡3 § 9 + 48 ) x= 4 p ¡3 § 57 ) x= 4 Solutions are: p ¡3 + 57 4

and

p ¡3 ¡ 57 : 4

EXERCISE 6A.3 1 Use the quadratic formula to solve exactly for x: b x2 + 6x + 7 = 0 a x2 ¡ 4x ¡ 3 = 0 e x2 ¡ 4x + 2 = 0 d x2 + 4x = 1 p h (3x + 1)2 = ¡2x g x2 ¡ 2 2x + 2 = 0

c f

x2 + 1 = 4x 2x2 ¡ 2x ¡ 3 = 0

i

(x + 3)(2x + 1) = 9

c

(x ¡ 2)2 = 1 + x

f

2x ¡

2 Use the quadratic formula to solve exactly for x: a

(x+2)(x¡1) = 2¡3x

b

(2x + 1)2 = 3 ¡ x

d

x¡1 = 2x + 1 2¡x

e

x¡

B

1 =1 x

1 =3 x

THE DISCRIMINANT OF A QUADRATIC

In the quadratic formula, the quantity b2 ¡ 4ac under the square root sign is called the discriminant.

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The symbol delta ¢ is used to represent the discriminant, so ¢ = b2 ¡ 4ac. p ¡b § ¢ The quadratic formula becomes x = where ¢ replaces b2 ¡ 4ac. 2a

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Note:

¡b x= is the only solution (a repeated or double root) 2a p ¢ is a positive real number, so there are two distinct real p p ¡b + ¢ ¡b ¡ ¢ roots: and 2a 2a p ¢ is not a real number and so there are no real roots.

²

If ¢ = 0,

²

If ¢ > 0,

²

If ¢ < 0,

²

If a, b and c are rational and ¢ is a perfect square then the equation has two rational roots which can be found by factorisation.

Example 7 Use the discriminant to determine the nature of the roots of: b 3x2 ¡ 4x ¡ 2 = 0 a 2x2 ¡ 2x + 3 = 0 a

¢ = b2 ¡ 4ac = (¡2)2 ¡ 4(2)(3) = ¡20 which is < 0 ) there are no real roots

¢ = b2 ¡ 4ac = (¡4)2 ¡ 4(3)(¡2) = 40 which is > 0 40 is not a perfect square so there are 2 distinct irrational roots.

b

Example 8 For x2 ¡ 2x + m = 0, find ¢ and hence find the values of m for which the equation has: a a repeated root b 2 distinct real roots c no real roots. x2 ¡ 2x + m = 0 has a = 1, b = ¡2 and c = m ) ¢ = b2 ¡ 4ac = (¡2)2 ¡ 4(1)(m) = 4 ¡ 4m a

b

For a repeated root ¢=0 ) 4 ¡ 4m = 0 ) 4 = 4m ) m=1

repeated

2 distinct real roots

Notice:

c

For 2 distinct real roots ¢>0 ) 4 ¡ 4m > 0 ) ¡4m > ¡4 ) m0 ¢=0 ¢>0 ¢0 ¢>0 ¢=0 ¢ ¡1: k 6 ¡9 or k > ¡1: k = ¡9 or k = ¡1: ¡9 < k < ¡1:

EXERCISE 6B 1 By using the discriminant only, state the nature of the solutions of: p b x2 + 2 3x + 3 = 0 c 3x2 + 2x ¡ 1 = 0 a x2 + 7x ¡ 3 = 0 e x2 + x + 5 = 0 f 16x2 ¡ 8x + 1 = 0 d 5x2 + 4x ¡ 3 = 0 2 By using the discriminant only, determine which of the following quadratic equations have rational roots which can be found by factorisation. a 6x2 ¡ 5x ¡ 6 = 0 b 2x2 ¡ 7x ¡ 5 = 0 c 3x2 + 4x + 1 = 0 2 2 e 4x ¡ 3x + 2 = 0 f 8x2 + 2x ¡ 3 = 0 d 6x ¡ 47x ¡ 8 = 0 3 For the following quadratic equations, determine ¢ in simplest form and draw a sign diagram for it. Hence find the value of m for which the equation has: i a repeated root ii two distinct real roots iii no real roots. 2 2 a x + 4x + m = 0 b mx + 3x + 2 = 0 c mx2 ¡ 3x + 1 = 0

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4 For the following quadratic equations, find the discriminant ¢ and hence draw a sign diagram for it. Find all k values for which the equation has: i two distinct real roots ii two real roots iii a repeated root iv no real roots. b kx2 ¡ 2x + k = 0 a 2x2 + kx ¡ k = 0 2 d 2x2 + [k ¡ 2]x + 2 = 0 c x + [k + 2]x + 4 = 0 f [k + 1]x2 + kx + k = 0 e x2 + [3k ¡ 1]x + [2k + 10] = 0

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C THE SUM AND PRODUCT OF THE ROOTS If ax2 + bx + c = 0 has roots ® and ¯ then ® + ¯ = ¡

b a

and ®¯ =

c . a

For example: if ® and ¯ are the roots of 2x2 ¡ 2x ¡ 1 = 0 then ® + ¯ = 1 and ®¯ = ¡ 12 . Proof: Method 1 (Quadratic formula) p p ¡b + ¢ ¡b ¡ ¢ Let ® = , ¯= 2a 2a p p ¡b + ¢ ¡ b ¡ ¢ ) ®+¯ = 2a

Method 2 As ax2 + bx + c = a(x ¡ ®)(x ¡ ¯), ax2 + bx + c = a(x2 ¡ [® + ¯]x + ®¯) b c ) x2 + x + = x2 ¡ [® + ¯]x + ®¯ a a

¡2b b =¡ 2a a Ã p ! p !Ã ¡b ¡ ¢ ¡b + ¢ and ®¯ = 2a 2a

Equating coefficients, b c ®+¯ =¡ and ®¯ = a a

=

=

b2 ¡ ¢ b2 ¡ (b2 ¡ 4ac) 4ac c = = 2 = 2 4a 4a2 4a a

Example 10 Find the sum and product of the roots of 25x2 ¡ 20x + 1 = 0. Check your answer by solving the quadratic. b 20 4 = = a 25 5 c 1 and ®¯ = = a 25

If ® and ¯ are the roots then ® + ¯ = ¡

Check: 25x2 ¡ 20x + 1 = 0 has roots p p p p 20 § 400 ¡ 4(25)(1) 20 § 300 20 § 10 3 2§ 3 = = = 50 50 50 5 p p 2+ 3 2¡ 3 4 These have sum = + = X 5 5 5 Ã p ! p !Ã 2¡ 3 4¡3 1 2+ 3 = = X and product = 5 5 25 25

EXERCISE 6C 1 Find the sum and product of the roots of:

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x2 + 11x ¡ 13 = 0

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3x2 ¡ 2x + 7 = 0

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5x2 ¡ 6x ¡ 14 = 0

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2 The equation kx2 ¡ (1 + k)x + (3k + 2) = 0 is such that the sum of its roots is twice their product. Find k and the two roots. 3 ax2 ¡ 6x + a ¡ 2 = 0, a 6= 0 has one root which is double the other. a Let the roots be ® and 2® and find two equations involving ®. b Find a and the two roots of the quadratic equation. 4 kx2 + (k ¡ 8)x + (1 ¡ k) = 0 has one root which is two more than the other. Find k and the two roots. 5 The roots of the equation x2 ¡ 6x + 7 = 0 are ® and ¯. 1 1 Find the simplest quadratic equation with roots ® + and ¯ + . ¯ ® 6 The roots of 2x2 ¡ 3x ¡ 5 = 0 are p and q. Find all quadratic equations with roots p2 + q and q2 + p. 7 kx2 + [k + 2]x ¡ 3 = 0 has roots which are real and positive. Find the possible values that k may have.

D

GRAPHING QUADRATIC FUNCTIONS

REVIEW OF TERMINOLOGY The equation of a quadratic function is given by y = ax2 + bx + c, where a 6= 0.

axis of symmetry

y

The graph of a quadratic function is called a parabola. The point where the graph ‘turns’ is called the vertex. If the graph opens upward, the y-coordinate of the vertex is the minimum and the graph is concave upwards.

parabola

x zero

If the graph opens downward, the y-coordinate of the vertex is the maximum and the graph is concave downwards.

zero y-intercept

minimum

The vertical line that passes through the vertex is called the axis of symmetry. All parabolas are symmetrical about the axis of symmetry.

vertex

The point where the graph crosses the y-axis is the y-intercept.

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The points (if they exist) where the graph crosses the x-axis are called the x-intercepts, and correspond to the zeros of the function.

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GRAPHING y = a(x¡®)(x¡¯)

INVESTIGATION 1

This investigation is best done using a graphing package or graphics calculator.

GRAPHING PACKAGE

TI C

What to do:

1

a Use technology to help you to sketch: y = (x ¡ 1)(x ¡ 3), y = 2(x ¡ 1)(x ¡ 3), y = ¡(x ¡ 1)(x ¡ 3), y = ¡3(x ¡ 1)(x ¡ 3) and y = ¡ 12 (x ¡ 1)(x ¡ 3) b Find the x-intercepts for each function in a. c What is the geometrical significance of a in y = a(x ¡ 1)(x ¡ 3)?

2

a Use technology to help you to sketch: y = 2(x ¡ 1)(x ¡ 4), y = 2(x ¡ 3)(x ¡ 5), y = 2x(x + 5) and y = 2(x + 2)(x + 4)

y = 2(x + 1)(x ¡ 2),

b Find the x-intercepts for each function in a. c What is the geometrical significance of ® and ¯ in y = 2(x ¡ ®)(x ¡ ¯)? 3

a Use technology to help you to sketch: y = 2(x ¡ 3)2 , y = 2(x ¡ 1)2 ,

y = 2(x + 2)2 ,

y = 2x2

b Find the x-intercepts for each function in a. c What is the geometrical significance of ® in y = 2(x ¡ ®)2 ? 4 Copy and complete: ² If a quadratic has factorisation y = a(x ¡ ®)(x ¡ ¯) it ...... the x-axis at ...... ² If a quadratic has factorisation y = a(x ¡ ®)2 it ...... the x-axis at ......

GRAPHING y = a(x¡h) 2+k

INVESTIGATION 2

This investigation is also best done using technology.

GRAPHING PACKAGE

What to do:

1

a Use technology to help you to sketch: y = (x ¡ 3)2 + 2, y = 2(x ¡ 3)2 + 2, y = ¡2(x ¡ 3)2 + 2, 1 2 and y = ¡ 3 (x ¡ 3)2 + 2 y = ¡(x ¡ 3) + 2 b Find the coordinates of the vertex for each function in a. c What is the geometrical significance of a in y = a(x ¡ 3)2 + 2?

2

a Use technology to help you to sketch: y = 2(x ¡ 2)2 + 4, y = 2(x ¡ 1)2 + 3, y = 2(x + 2)2 ¡ 5 y = 2(x + 1)2 + 4,

TI C

y = 2(x ¡ 3)2 + 1, and y = 2(x + 3)2 ¡ 2

b Find the coordinates of the vertex for each function in a. c What is the geometrical significance of h and k in y = 2(x ¡ h)2 + k?

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then its vertex has coordinates ......

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3 Copy and complete: If a quadratic is in the form y = a(x ¡ h)2 + k

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From Investigations 1 and 2 you should have discovered that: The coefficient of x2 (which is a) controls the degree of width of the graph and whether it opens upwards or downwards.

²

I a > 0 produces

or concave up. a < 0 produces

or concave down.

I If ¡1 < a < 1, a 6= 0 the graph is wider than y = x2 : If a < ¡1 or a > 1 the graph is narrower than y = x2 :

Quadratic form, a 6= 0

Graph

² y = a(x ¡ ®)(x ¡ ¯) ®, ¯ are real

Facts x-intercepts are ® and ¯

a

x

b x=

² y = a(x ¡ ®)2 ® is real

a+b 2

x=a

touches x-axis at ® axis of symmetry is x = ® vertex is (®, 0) x

V (a, 0)

² y = a(x ¡ h)2 + k

axis of symmetry is x = ®+¯ 2 ³ ´ ®+¯ ®+¯ vertex is 2 , f( 2 )

axis of symmetry is x = h vertex is (h, k)

x=h

V (h, k)

² y = ax2 + bx + c (general quadratic form)

p -b - ¢ 2a

p x -b + ¢ 2a

x=

-b 2a

axis of symmetry is ¡b x= 2a x-intercepts for ¢ > 0 are p ¡b § ¢ 2a where ¢ = b2 ¡ 4ac

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p p b ¡b ¡ ¢ ¡b + ¢ Note: ¡ is the average of and irrespective of the sign of ¢. 2a 2a 2a

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SKETCHING GRAPHS USING KEY FACTS Example 11 Using axes intercepts only, sketch the graphs of: a y = 2(x + 3)(x ¡ 1) b y = ¡2(x ¡ 1)(x ¡ 2) a

y = 2(x + 3)(x ¡ 1) has x-intercepts ¡3, 1

y = ¡2(x ¡ 1)(x ¡ 2) has x-intercepts 1, 2

b

When x = 0, y = 2(3)(¡1) = ¡6

When x = 0, y = ¡2(¡1)(¡2) = ¡4

) y-intercept is ¡6

) y-intercept is ¡4

y

y = 12 (x + 2)2

c

y = 12 (x + 2)2 touches x-axis at ¡2 When x = 0, y = 12 (2)2 =2 ) y-intercept is 2 y

y 1 2

-3

c

x 2

x

1

-4 -2

-6

x

Example 12 Use the vertex, axis of symmetry and y-intercept to graph y = ¡2(x + 1)2 + 4: The vertex is (¡1, 4).

V(-1, 4)

y

The axis of symmetry is x = ¡1: 2

When x = 0, y = ¡2(1)2 + 4 =2 )

a0 )

shape

When x = ¡ 32 , y = 2(¡ 32 )2 + 6(¡ 32 ) ¡ 3

) axis of symmetry is x = ¡ 32 .

= ¡7 12

fsimplifyingg

) vertex is (¡ 32 , ¡7 12 ). When x = 0, y = ¡3 ) y-intercept is ¡3.

c

d

When y = 0, 2x2 + 6x ¡ 3 = 0 p ¡6 § 36 ¡ 4(2)(¡3) ) x= 4 ) x ¼ ¡3:44 or 0:44

x =- 23

y

x » -3.44

-3

» 0.44

V(-1 21 , - 7 21 )

Example 14 Determine the coordinates of the vertex of y = 2x2 ¡ 8x + 1: y = 2x2 ¡ 8x + 1 has a = 2, b = ¡8, c = 1 ¡b ¡(¡8) and so = =2 2a 2£2

The vertex is called the maximum turning point or the minimum turning point depending on whether the graph is concave down or concave up.

) equation of axis of symmetry is x = 2 and when x = 2, y = 2(2)2 ¡ 8(2) + 1 = ¡7 ) the vertex has coordinates (2, ¡7).

EXERCISE 6D.1 1 Using axes intercepts only, sketch the graphs of: a y = (x ¡ 4)(x + 2) b y = ¡(x ¡ 4)(x + 2) d y = ¡3x(x + 4) e y = 2(x + 3)2

c f

y = 2(x + 3)(x + 5) y = ¡ 14 (x + 2)2

c f

y = ¡2(x ¡ 1)2 ¡ 3 1 y = ¡ 10 (x + 2)2 ¡ 3

2 What is the axis of symmetry of each graph in question 1? 3 Use the vertex, axis of symmetry and y-intercept to graph: b y = 2(x + 2)2 + 1 a y = (x ¡ 1)2 + 3 1 2 e y = ¡ 13 (x ¡ 1)2 + 4 d y = 2 (x ¡ 3) + 2

4 Find the turning point or vertex for the following quadratic functions:

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y = x2 + 2x ¡ 3 y = 2x2 + 8x ¡ 7 y = 2x2 ¡ 10x + 3

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y = x2 ¡ 4x + 2 y = ¡3x2 + 1 y = 2x2 + 6x ¡ 1

a d g

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c f i

y = 2x2 + 4 y = ¡x2 ¡ 4x ¡ 9 y = ¡ 12 x2 + x ¡ 5

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5 Find a d g j

the x-intercepts for: y = x2 ¡ 9 y = x2 + x ¡ 12 y = ¡2x2 ¡ 4x ¡ 2 y = x2 + 4x ¡ 3

b e h k

y y y y

= 2x2 ¡ 6 = 4x ¡ x2 = 4x2 ¡ 24x + 36 = x2 ¡ 6x ¡ 2

6 For the following quadratics, find: i the equation of the axis of symmetry iii the axes intercepts, if they exist. a d g

y = x2 ¡ 2x + 5 y = ¡x2 + 3x ¡ 2 y = 6x ¡ x2

ii iv

y y y y

= x2 + 7x + 10 = ¡x2 ¡ 6x ¡ 8 = x2 ¡ 4x + 1 = x2 + 8x + 11

the coordinates of the vertex Hence, sketch the graph.

y = x2 + 4x ¡ 1 y = ¡3x2 + 4x ¡ 1 y = ¡x2 ¡ 6x ¡ 8

b e h

c f i l

c f i

y = 2x2 ¡ 5x + 2 y = ¡2x2 + x + 1 y = ¡ 14 x2 + 2x + 1

SKETCHING GRAPHS BY COMPLETING THE SQUARE If we wish to find the vertex of a quadratic given in general form y = ax2 + bx + c then one approach is to convert it to the form y = a(x ¡ h)2 + k where we can read off the vertex (h, k). To do this we may choose to ‘complete the square’.

Example 15 Write y = x2 + 4x + 3 in the form y = (x ¡ h)2 + k by completing the square. Hence sketch y = x2 + 4x + 3, stating the coordinates of the vertex. y = x2 + 4x + 3 y = x2 + 4x + 22 + 3 ¡ 22 y = (x + 2)2 ¡ 1

) )

shift 2 units left

-1

shift 1 unit down

-1 @\=\!X\+\4!\+3

Vertex is (¡2, ¡1) and y-intercept is 3.

-2

@\=\!X

-2 vertex (-2'\-1)

Example 16 Convert y = 3x2 ¡ 4x + 1 into the form y = a(x ¡ h)2 + k by ‘completing the square’. Hence, write down the coordinates of its vertex and sketch the graph of the function. y = 3x2 ¡ 4x + 1 = 3[x2 ¡ 43 x + 13 ]

ftake out a factor of 3g

= 3[x2 ¡ 2( 23 )x + ( 23 )2 ¡ ( 23 )2 + 13 ]

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= 3[(x ¡ 23 )2 ¡ 19 ] = 3(x ¡

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fexpand to desired formg

So the vertex is ( 23 , ¡ 13 ). 1

The y-intercept is 1.

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!

1 1.5 V(We_ ,-Qe_)

We can use technology to confirm this. For example:

EXERCISE 6D.2 1 Write the following quadratics in the form y = (x ¡ h)2 + k by ‘completing the square’. Hence sketch each function, stating the vertex: a y = x2 ¡ 2x + 3 b y = x2 + 4x ¡ 2 c y = x2 ¡ 4x e y = x2 + 5x ¡ 2 f y = x2 ¡ 3x + 2 d y = x2 + 3x h y = x2 + 8x ¡ 2 i y = x2 ¡ 5x + 1 g y = x2 ¡ 6x + 5 2 For each i ii iii iv v

of the following quadratics: convert into the form y = a(x ¡ h)2 + k state the coordinates of the vertex find the y-intercept. Hence, sketch the graph of the quadratic. Use technology to check your answer.

y = 2x2 + 4x + 5 y = 2x2 ¡ 6x + 1 y = ¡x2 + 4x + 2

a c e

b d f

by ‘completing the square’ a is always the factor to be ‘taken out’.

y = 2x2 ¡ 8x + 3 y = 3x2 ¡ 6x + 5 y = ¡2x2 ¡ 5x + 3

3 Use your graphing package or graphics calculator to graph each of the following functions. Hence write each function in the form y¡=¡a(x¡¡¡h)2¡+¡k. y = x2 ¡ 4x + 7 y = 2x2 + 6x ¡ 4

a d

b e

y = x2 + 6x + 3 y = ¡2x2 ¡ 10x + 1

GRAPHING PACKAGE

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c f

y = ¡x2 + 4x + 5 y = 3x2 ¡ 9x ¡ 5

THE DISCRIMINANT AND THE QUADRATIC GRAPH

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Consider the graphs of: y = x2 ¡ 2x + 3, y = x2 ¡ 2x + 1, y = x2 ¡ 2x ¡ 3. All of these curves have the same axis of symmetry: x = 1.

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y = x2 ¡ 2x + 3

y = x2 ¡ 2x + 1

y

y = x2 ¡ 2x ¡ 3

y

-1

1

3 1

¢ = b2 ¡ 4ac = (¡2)2 ¡ 4(1)(3) = ¡8

¢ = b2 ¡ 4ac = (¡2)2 ¡ 4(1)(1) =0

¢0

² ² ²

The discriminant ¢ determines if the graph:

x

3

-3

x

1

x

1

cuts the x-axis twice

does not cut the x-axis touches the x-axis cuts the x-axis twice

(¢ < 0) (¢ = 0) (¢ > 0).

Example 17 Use the discriminant to determine the relationship between the graph and b y = ¡2x2 + 5x + 1 the x-axis for: a y = x2 + 3x + 4 a

b

a = 1, b = 3, c = 4 ) ¢ = b2 ¡ 4ac = 9 ¡ 4(1)(4) = ¡7 which is < 0 The graph does not cut the x-axis.

a > 0 ) concave up x

It lies entirely above the x-axis.

a = ¡2, b = 5, c = 1 ) ¢ = b2 ¡ 4ac = 25 ¡ 4(¡2)(1) = 33 which is > 0

a < 0 ) concave down

x

) the graph cuts the x-axis twice.

POSITIVE DEFINITE AND NEGATIVE DEFINITE QUADRATICS Positive definite quadratics are quadratics which are positive for all values of x, i.e., ax2 + bx + c > 0 for all x 2 R . Negative definite quadratics are quadratics which are negative for all values of x, i.e., ax2 + bx + c < 0 for all x 2 R .

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² A quadratic is positive definite if a > 0 and ¢ < 0. ² A quadratic is negative definite if a < 0 and ¢ < 0.

Tests:

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EXERCISE 6D.3 1 Use the discriminant to determine the relationship between the graph and x-axis for: p b y = x2 + 4 2x + 8 c y = ¡2x2 + 3x + 1 a y = x2 + 7x ¡ 2 e y = ¡x2 + x + 6 f y = 9x2 + 6x + 1 d y = 6x2 + 5x ¡ 4 2 Show that: a x2 ¡ 3x + 6 > 0 for all x c 2x2 ¡ 4x + 7 is positive definite

b d

4x ¡ x2 ¡ 6 < 0 for all x ¡2x2 + 3x ¡ 4 is negative definite

3 Explain why 3x2 + kx ¡ 1 is never positive definite for any value of k. 4 Under what conditions is 2x2 + kx + 2 positive definite?

E FINDING A QUADRATIC FROM ITS GRAPH If we are given sufficient information on or about a graph we can determine the quadratic function in whatever form is required.

Example 18 Find the equation of the quadratic with graph: a b y

y

3

8 3

-1

x 2

a

Since the x-intercepts are ¡1 and 3, y = a(x + 1)(x ¡ 3), a < 0. But when x = 0, y = 3 ) 3 = a(1)(¡3) ) a = ¡1 So, y = ¡(x + 1)(x ¡ 3):

b

x

Since it touches at 2, y = a(x ¡ 2)2 , a > 0. But when x = 0, y = 8 ) 8 = a(¡2)2 ) a=2 So, y = 2(x ¡ 2)2 :

Example 19 The axis of symmetry is x = 1, so the other x-intercept is 4

Find the equation of the quadratic with graph:

)

y 16

But when x = 0, y = 16 ) 16 = a(2)(¡4) ) a = ¡2

x

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y = a(x + 2)(x ¡ 4)

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Example 20 Find, in the form y = ax2 + bx + c, the equation of the quadratic whose graph cuts the x-axis at 4 and ¡3 and passes through the point (2, ¡20). Since the x-intercepts are 4 and ¡3, the equation is y = a(x ¡ 4)(x + 3) where a 6= 0. But when x = 2, y = ¡20 ) ¡20 = a(2 ¡ 4)(2 + 3) ) ¡20 = a(¡2)(5) ) a=2 ) the equation is y = 2(x ¡ 4)(x + 3), or y = 2x2 ¡ 2x ¡ 24

Example 21 a

b

y

Find the equation of the quadratic given its graph is:

y

V(-4, 2)

16

x

-2 x V(3,-2)

For a vertex (3, ¡2) the quadratic has the form y = a(x ¡ 3)2 ¡ 2 But when x = 0, y = 16 ) 16 = a(¡3)2 ¡ 2 ) 16 = 9a ¡ 2 ) 9a = 18 ) a=2 So, y = 2(x ¡ 3)2 ¡ 2

a

b

For a vertex (¡4, 2) the quadratic has the form y = a(x + 4)2 + 2 But when x = ¡2, y = 0 ) 0 = a(2)2 + 2 ) 4a = ¡2 ) a = ¡ 12 So, y = ¡ 12 (x + 4)2 + 2

EXERCISE 6E 1 Find the equation of the quadratic with graph: a b y y

c

8

4 1

d

3

x

2

x

2

e

y

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2 Find the quadratic with graph: a b y

c

y

y

4

12

x

-4

-12

x

x

2

x =-3

x =-1

x=3

3 Find, in the form y = ax2 + bx + c, the equation of the quadratic whose graph: cuts the x-axis at 5 and 1, and passes through (2, ¡9) cuts the x-axis at 2 and ¡ 12 , and passes through (3, ¡14) touches the x-axis at 3 and passes through (¡2, ¡25) touches the x-axis at ¡2 and passes through (¡1, 4) cuts the x-axis at 3, passes through (5, 12) and has axis of symmetry x = 2 cuts the x-axis at 5, passes through (2, 5) and has axis of symmetry x = 1.

a b c d e f

4 If V is the vertex, find the equation of the quadratic given its graph is: y y y a b c

V(3,¡8)

V(2,¡4)

x

7

x

1 x V(2,-1)

y

d

y

e 7

&Ew_ ' Qw_*

(3,¡1)

x

y

f

V(2,¡3) x

x

V(4,-6)

V&Qw_ '-\wwE_*

INVESTIGATION 3

FINDING QUADRATIC FUNCTIONS

y¡=¡2x2¡+¡3x¡+¡7 is a quadratic function from which the following table of values is obtained:

x y

0 7

1 12

2 21

3 34

4 51

5 72

Consider adding two further rows to this table: a row called ¢1 which gives differences between successive y-values, and a row called ¢2 which gives differences between successive ¢1 -values. 2 21

5

9 4

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1 12

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0 7

50

x y

75

So, we have:

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This table is known as a difference table. What to do:

1 Construct difference tables (for x = 0, 1, 2, 3, 4, 5) for each of the following quadratic functions: a y = x2 + 4x + 3 b y = 3x2 ¡ 4x c y = 5x ¡ x2 d y = 4x2 ¡ 5x + 2 2 What do you notice about the ¢2 row for the quadratic functions in 1? 3 Consider the general quadratic y = ax2 + bx + c, a 6= 0. a Copy and complete the following difference table: x 0 1 2 3 4 5 y ° c a+b+c 4a + 2b + c :::::: :::::: :::::: ° :::::: :::::: :::::: :::::: ¢1 ° :::::: :::::: :::::: ¢2 b Comment on the ¢2 row. c What can the encircled numbers be used for? 4 Use what you have noticed in 3 to determine, if possible, the quadratic functions with the following tables of values: a

x y

0 6

1 5

2 8

c

x y

0 1

1 2

2 ¡1

3 15

4 26

3 ¡8

4 ¡18

b

x y

0 8

1 10

d

x y

0 5

1 3

2 18

3 32

4 52

2 ¡1

3 ¡7

4 ¡15

5 Cutting up Pizzas Given a pizza, we wish to determine the maximum number of pieces into which it can be cut using n cuts across it. For example, for n = 1

we have

i.e., 2 pieces

1 2

for n = 3

1

6

we have

5

a Copy and complete: Number of cuts, n Maximum number of pieces, Pn

i.e., 7 pieces

2

7 4

3

0

1

2

3

4

5

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b Complete ¢1 and ¢2 rows and hence determine (if possible) a quadratic formula for Pn . c For a huge pizza with 12 cuts across it, find the maximum number of pieces resulting.

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F

WHERE FUNCTIONS MEET

Consider the graphs of a quadratic function and a linear function on the same set of axes. Notice that we could have:

cutting (2 points of intersection)

touching (1 point of intersection)

missing (no points of intersection)

If the graphs meet, the coordinates of the points of intersection of the graphs of the two functions can be found by solving the two equations simultaneously.

Example 22 Find the coordinates of the points of intersection of the graphs with equations y = x2 ¡ x ¡ 18 and y = x ¡ 3: y = x2 ¡ x ¡ 18 meets y = x ¡ 3 where x2 ¡ x ¡ 18 = x ¡ 3 fRHS = 0g ) x2 ¡ 2x ¡ 15 = 0 ) (x ¡ 5)(x + 3) = 0 ffactorisingg ) x = 5 or ¡3 Substituting into y = x ¡ 3, when x = 5, y = 2 and when x = ¡3, y = ¡6. ) the graphs meet at (5, 2) and (¡3, ¡6).

Example 23 y = 2x + c is a tangent to y = 2x2 ¡ 3x + 4. Find c. y = 2x + c meets y = 2x2 ¡ 3x + 4 where 2x2 ¡ 3x + 4 = 2x + c 2 ) 2x ¡ 5x + (4 ¡ c) = 0 Now this quadratic has ¢ = 0 since the graphs touch. ) (¡5)2 ¡ 4(2)(4 ¡ c) = 0 ) 8(4 ¡ c) = 25 ) 4 ¡ c = 3 18

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EXERCISE 6F 1 Find the coordinates of the point(s) of intersection of the graphs with equations: b y = ¡x2 + 3x + 9 and y = 2x ¡ 3 a y = x2 ¡ 2x + 8 and y = x + 6 c y = x2 ¡ 4x + 3 and y = 2x ¡ 6 d y = ¡x2 + 4x ¡ 7 and y = 5x ¡ 4 2 Use a graphing package or a GDC to find the coordinates of the points of intersection (to 2 decimal places) of the graphs with equations: a c

GRAPHING PACKAGE

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y = x2 ¡ 3x + 7 and y = x + 5 y = ¡x2 ¡ 2x + 4 and y = x + 8

y = x2 ¡ 5x + 2 and y = x ¡ 7 y = ¡x2 + 4x ¡ 2 and y = 5x ¡ 6

b d

3 Find, by algebraic means, the points of intersection of the graphs with equations: b y = x2 + 2x ¡ 3 and y = x ¡ 1 a y = x2 and y = x + 2 c y = 2x2 ¡ x + 3 and y = 2 + x + x2 d xy = 4 and y = x + 3 4 Use technology to check your solutions to the questions in 3. 5 Find possible values of c for which the lines y = 3x + c are tangents to the parabola with equation y = x2 ¡ 5x + 7: 6 Find the values of m for which the lines y = mx ¡ 2 are tangents to the curve with equation y = x2 ¡ 4x + 2: 7 Find the slopes of the lines with y-intercept (0, 1) that are tangents to the curve y = 3x2 + 5x + 4. 8

a For what values of c do the lines y = x + c never meet the parabola with equation y = 2x2 ¡ 3x ¡ 7? b Choose one of the values of c found in part a above and sketch the graphs using technology to illustrate that these curves never meet.

THE PARABOLA y2 = 4ax

INVESTIGATION 4 y

A parabola is defined as the locus of all points which are equidistant from a fixed point called the focus and a fixed line called the directrix.

P(x,¡y)

N

x

Suppose the focus is F(a, 0) and the directrix is the vertical line x = ¡a.

F(a,¡0) x¡=¡-a

What to do:

1 Suggest a reason or reasons why we would let the focus be at (a, 0) and the directrix be the line x¡=¡¡a.

PRINTABLE GRAPH PAPER

2 Use the circular-linear graph paper provided to graph the parabola which has focus F(2, 0) and directrix x¡=¡¡2.

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3 Use the definition given above to show that the equation of the parabola is y2 = 4ax.

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4 Let y = mx + c be a tangent to y 2 = 4ax y at the point P. Use quadratic theory to show that: a a = mc µ ¶ y¡=¡mx¡+¡c a 2a , . b P is at 2 m m 5 Suppose a ray of light comes in parallel to the axis of symmetry (the x-axis). It strikes a parabolic mirror at P and is reflected to cut the x-axis at R(k, 0). ®1 = ®2 by the Reflection Principle.

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P

x

y az

P ax Q

R

x

a Deduce that triangle PQR is isosceles. b Hence, deduce that k = a. 6 Clearly state what special result follows from 5 b. 7 List real life examples of where the result in 6 has been utilised.

G PROBLEM SOLVING WITH QUADRATICS When solving some problems algebraically, a quadratic equation results. We are generally only interested in any real solutions which result. If the resulting quadratic equation has no real roots then the problem has no real solution. Any answer we obtain must be checked to see if it is reasonable. For example: ² if we are finding a length then it must be positive and we reject any negative solutions ² if we are finding ‘how many people are present’ then clearly a fractional answer would be unacceptable. General problem solving method: Step 1: Step 2: Step 3: Step 4:

If the information is given in words, translate it into algebra using a pronumeral such as x for the unknown. Write down the resulting equation. Solve the equation by a suitable method. Examine the solutions carefully to see if they are acceptable. Give your answer in a sentence.

Example 24 A rectangle has length 3 cm longer than its width. Its area is 42 cm2 . Find its width.

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If the width is x cm, then the length is (x + 3) cm. x cm Therefore x(x + 3) = 42 fequating areasg 2 ) x + 3x ¡ 42 = 0 (x + 3) cm ) x ¼ ¡8:15 or 5:15 fusing technologyg We reject the negative solution as lengths are positive ) width ¼ 5:15 cm.

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Example 25 Is it possible to bend a 12 cm length of wire to form the legs of a right angled triangle with area 20 cm2 ?

becomes

12 cm

1 2 x(12

Area, A = x) = 20 ) ) x(12 ¡ x) = 40 ) 12x ¡ x2 ¡ 40 = 0 1 2 x(12 ¡

area 20 cm 2

¡ x) x

12 - x

p ¡16 ) x ¡ 12x + 40 = 0 which gives x = 2 There are no real solutions, indicating the impossibility. 12 §

2

Example 26 A wall is 12 m long. It is timber panelled using vertical sheets of panelling of equal width. If the sheets had been 0:2 m wider, 2 less sheets would have been required. What is the width of the timber panelling used? Let x m be the width of each panel. 12 ) is the number of sheets needed. x ¢ ¡ Now if the sheets are x + 15 m in width µ ¶ 12 ¡ 2 sheets are needed. x

...... xm 12 m

¶ 12 ¡ 2 = 12 x 12 2 ¡ = 12 ) 12 ¡ 2x + 5x 5 12 2 ¡ =0 ) ¡2x + 5x 5 ) ¡10x2 + 12 ¡ 2x = 0 ) 5x2 + x ¡ 6 = 0 ) (5x + 6)(x ¡ 1) = 0 ) x = ¡ 65 or 1 ) each sheet is 1 m wide. ¢ ¡ So, x + 15

µ

flength of wallg fexpanding LHSg

f£ each term by 5xg f¥ each term by ¡2g where x > 0

EXERCISE 6G

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1 Two integers differ by 12 and the sum of their squares is 74. Find the integers.

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2 The sum of a number and its reciprocal is 5 15 . Find the number. 3 The sum of a natural number and its square is 210. Find the number. 4 The product of two consecutive even numbers is 360. Find the numbers. 5 The product of two consecutive odd numbers is 255. Find the numbers. n 6 The number of diagonals of an n-sided polygon is given by the formula D = (n¡3). 2 A polygon has 90 diagonals. How many sides does it have? 7 The length of a rectangle is 4 cm longer than its width. Find its width given that its area is 26 cm2 . 8 A rectangular box has a square base, and its height is 1 cm longer than the length of each side of its base. a If each side of its base has length x cm, show that its total surface area is given by A = 6x2 + 4x cm2 . b If the total surface area is 240 cm2 , find the dimensions of the box.

x cm

An open box contains 80 cm3 . It is made from a square piece of tinplate with 3 cm squares cut from each of its 4 corners. Find the dimensions of the original piece of tinplate.

9

3 cm

10 Is it possible to bend a 20 cm length of wire into the shape of a rectangle which has an area of 30 cm2 ? 11 The golden rectangle is the rectangle defined by the following statement: The golden rectangle can be divided into a square and a smaller rectangle by a line which is parallel to its shorter sides, the smaller rectangle being similar to the original rectangle.

A

Y

B

D

X

C

Thus, if ABCD is the golden rectangle, ADXY is a square and BCXY is similar to ABCD. AB The ratio of for the golden rectangle is called the golden ratio. AD p 1+ 5 . Hint: Let AB = x units and AD = 1 unit. Show that the golden ratio is 2 12

A triangular paddock has a road AB forming its longest side. AB is 3 km long. The fences AC and CB are at right angles. If BC is 400 m longer than AC, find the area of the paddock in hectares.

A 3 km B

C

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13 Find the width of a uniform concrete path placed around a 30 m by 40 m rectangular lawn, given that the concrete has area one quarter of the lawn.

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14 Chuong and Hassan both drive 40 km from home to work each day. One day Chuong said to Hassan, “If you drive home at your usual speed, I will average 40 km h¡1 faster than you and arrive home in 20 minutes less time.” Find Hassan’s speed. 15 If the average speed of a small aeroplane had been 120 km h¡1 less, it would have taken a half an hour longer to fly 1000 km. Find the speed of the plane. 16 Two trains travel a 160 km track each day. The express travels 10 km h¡1 faster and takes 30 minutes less than the normal train. Find the speed of the express. 17

A group of elderly citizens chartered a bus for $160. However, at the last minute 8 of them fell ill and had to miss the trip. As a consequence, the other citizens had to pay an extra $1 each. How many elderly citizens went on the trip?

18 A tunnel is parabolic in shape with dimensions shown: A truck carrying a wide load is 4:8 m high and 3:9 m wide, and needs to pass through the tunnel. Determine whether the truck will fit.

8m

6m parabolic arch vertical supports

B

19

AB is the longest vertical support of a bridge which contains a parabolic arch. The vertical supports are 10 m apart. The arch meets the vertical end supports 6 m above the road.

70 m 6¡m 10 m

A

roadway

a If axes are drawn on the diagram of the bridge above, with x-axis the road and y-axis on AB, find the equation of the parabolic arch in the form y = ax2 + c. b Hence, determine the lengths of all other vertical supports.

H

QUADRATIC OPTIMISATION

For a quadratic function y = ax2 + bx + c, we have already seen that:

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if a < 0, the maximum value of y occurs b at x = ¡ . 2a

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value of y occurs b at x = ¡ 2a

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The process of finding the maximum or minimum value of a function is called optimisation. ² ²

Optimisation is a very useful tool when looking at such issues as:

maximising profits minimising costs

Example 27 Find the maximum or minimum value of the following quadratics, and the b y = 3 + 3x ¡ 2x2 corresponding value of x: a y = x2 + x ¡ 3 a

For y = x2 + x ¡ 3 a = 1, b = 1, c = ¡3:

For y = ¡2x2 + 3x + 3 a = ¡2, b = 3, c = 3:

b

As a > 0, shape is

As a < 0, shape is

) the minimum value occurs ¡b when x = = ¡ 12 2a

) the maximum value occurs ¡b ¡3 when x = = = 34 2a ¡4

and y = (¡ 12 )2 + (¡ 12 ) ¡ 3

and y = ¡2( 34 )2 + 3( 34 ) + 3

= ¡3 14

= 4 18

) the minimum value of y is ¡3 14

) the maximum value of y is 4 18

occurring when x = ¡ 12 :

occurring when x = 34 :

Example 28 A vegetable gardener has 40 m of fencing to enclose a rectangular garden plot where one side is an existing brick wall. If the two equal sides are x m long: a show that the area enclosed is given by A = x(40 ¡ 2x) m2 b find the dimensions of the vegetable garden of maximum area. xm

X

Y

xm

a

Side XY = 40 ¡ 2x m. Now area = length £ width ) A = x(40 ¡ 2x) m2 :

b

A = 40x ¡ 2x2 = ¡2x2 + 40x is a quadratic in x, with a = ¡2, b = 40, c = 0: As a < 0, shape is

Z

xm

brick wall

¡b ¡40 = = 10 2a ¡4 ) the area is maximised when YZ = 10 m and XY = 20 m.

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The max. area occurs when x =

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EXERCISE 6H 1 Find the maximum or minimum values of the following quadratics, and the corresponding values of x: a y = x2 ¡ 2x b y = 7 ¡ 2x ¡ x2 c y = 8 + 2x ¡ 3x2 d y = 2x2 + x ¡ 1 e y = 4x2 ¡ x + 5 f y = 7x ¡ 2x2 2 The profit in manufacturing x refrigerators per day, is given by the profit relation P = ¡3x2 + 240x ¡ 800 dollars. How many refrigerators should be made each day to maximise the total profit? What is the maximum profit? 3 A rectangular plot is enclosed by 200 m of fencing and has an area of A square metres. Show that: a A = 100x ¡ x2 where x m is the length of one of its sides b the area is maximised when the rectangle is a square.

xm

4 A rectangular paddock to enclose horses is to be made with one side being a straight water drain. If 1000 m of fencing is available for the other 3 sides, what dimensions should be used for the paddock so that it encloses the maximum possible area? ym

5 1800 m of fencing is available to fence six identical pig pens as shown in the diagram. a Explain why 9x + 8y = 1800: b Show that the total area of each pen is given by A = ¡ 98 x2 + 225x m2 .

xm

c If the area enclosed is to be maximised, what is the shape of each pen? 6 If 500 m of fencing is available to make 4 rectangular pens of identical shape, find the dimensions that maximise the area of each pen if the plan is: y

7

y = xX - 3x

2

b

The graphs of y = x2 ¡ 3x and y = 2x ¡ x2 are illustrated. a Prove that the graphs meet where x = 0 and x = 2 12 . b Find the maximum vertical separation between the curves for 0 6 x 6 2 12 .

x

3

a

y = 2x - xX

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8 Infinitely many rectangles may be inscribed B A within the right angled triangle shown alongside. 6 cm One of them is illustrated. a Let AB = x cm and BC = y cm. D C Use similar triangles to find y in terms of x. 8 cm b Find the dimensions of rectangle ABCD of maximum area.

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9 A manufacturer of pot-belly stoves has the following situation to consider. If x are made per week, each one will cost (50 + 400 x ) dollars and the total receipts 2 per week for selling them would be (550x ¡ 2x ) dollars. How many pot-belly stoves should be made per week in order to maximise profits? ¡1 2 ¢ 10 The total cost of producing x toasters per day is given by C = 10 x + 20x + 25 euros, and the selling price of each toaster is (44 ¡ 15 x) euros. How many toasters should be produced each day in order to maximise the total profit? 11 A manufacturer of barbeques knows that if x of them are made each week then each one will cost (60 +

800 x )

pounds and the total receipts per week will be (1000x ¡ 3x2 )

pounds. How many barbeques should be made per week to maximise profits? 12

The points P1 (a1 , b1 ), P2 (a2 , b2 ), P3 (a3 , b3 ), ......, Pn (an , bn ) are obtained by experiment. These points are believed to be close to linear through the origin O(0, 0). To find the equation of the ‘line of best fit’ through the origin we decide to minimise

y Mn Pc Mv Pz

Mx

Mc

Pn Pv

Px

Mz

(P1 M1 )2 +(P2 M2 )2 +(P3 M3 )2 +::::::+(Pn Mn )2 where [ Pi Mi ] is the vertical line segment connecting each point Pi with the line.

x

Find the slope of the ‘line of best fit’ in terms of ai and bi (i = 1, 2, 3, 4, ......, n). 13 Write f (x) = (x ¡ a ¡ b)(x ¡ a + b)(x + a ¡ b)(x + a + b) in expanded form and hence determine the least value of f (x). Assume that a and b are real constants. 14 By considering f (x) = (a1 x ¡ b1 )2 + (a2 x ¡ b2 )2 , use quadratic theory to prove the p p Cauchy-Schwarz inequality: j a1 b1 + a2 b2 j 6 a 2 + a 2 b 2 + b 2 . 1

2

1

2

15 b1 , c1 , b2 and c2 are real numbers such that b1 b2 = 2(c1 + c2 ). Show that at least one of the equations x2 + b1 x + c1 = 0, x2 + b2 x + c2 = 0 has two real roots.

REVIEW SET 6A

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b state the equation of the axis of symmetry d find the y-intercept f use technology to check your answers.

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y = 12 (x ¡ 2)2 ¡ 4: state the equation of the axis of symmetry find the coordinates of the vertex c find the y-intercept sketch the graph of the function e use technology to check your answers.

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2 For a b d

y = ¡2(x + 2)(x ¡ 1): state the x-intercepts find the coordinates of the vertex sketch the graph of the function

75

1 For a c e

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3 For a b d

y = 2x2 + 6x ¡ 3: convert into the form y = (x ¡ h)2 + k by ‘completing the square’ state the coordinates of the vertex c find the y-intercept sketch the graph e use technology to check your answers.

4 Solve the following equations, giving exact answers: a x2 ¡ 11x = 60 b 3x2 ¡ x ¡ 10 = 0

c

3x2 ¡ 12x = 0

c

2x2 ¡ 7x + 3 = 0

5 Solve the following equations:

x2 + 10 = 7x

a

b

x+

12 =7 x

6 Solve the following equation by completing the square: x2 + 7x ¡ 4 = 0 7 Solve the following using the quadratic formula: a x2 ¡ 7x + 3 = 0 b 2x2 ¡ 5x + 4 = 0 8

a For what values of c do the lines with equations y = 3x + c intersect the parabola y = x2 + x ¡ 5 in two distinct points? b Choose one such value of c from part a and find the points of intersection.

9 The roots of 2x2 ¡ 3x = 4 are ® and ¯. Find the simplest quadratic equation 1 1 which has roots and . ¯ ®

REVIEW SET 6B 1 Draw the graph of y = ¡x2 + 2x. 2 Find the equation of the axis of symmetry and the vertex of y = ¡3x2 + 8x + 7: 3 Use the discriminant only to determine the number of solutions to: a 3x2 ¡ 5x + 7 = 0 b ¡2x2 ¡ 4x + 3 = 0 4 Find the maximum or minimum value of the relation y = ¡2x2 + 4x + 3 and the value of x for which the maximum or minimum occurs. 5 Find the points of intersection of y = x2 ¡ 3x and y = 3x2 ¡ 5x ¡ 24. 6 For what values of k does the graph of y = ¡2x2 + 5x + k not cut the x-axis? C

D

existing wall

7 60 m of chicken wire is available to construct a chicken enclosure against an existing wall. The enclosure is to be rectangular. a If BC = x m, show that the area of rectangle ABCD is given by A = (30x ¡ 12 x2 ) m2 . b Find the dimensions of the enclosure which will maximise the area enclosed.

xm

B

A

8 For what values of m are the lines y = mx ¡ 10 tangents to the parabola y = 3x2 + 7x + 2?

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9 One of the roots of kx2 + (1 ¡ 3k)x + (k ¡ 6) = 0 is the negative reciprocal of the other root. Find k and the two roots.

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REVIEW SET 6C x2 + 5x + 3 = 0

1 Solve using the quadratic formula: a

x2 ¡ 5x ¡ 3 = 0

3x2 + 11x ¡ 2 = 0

b

2x2 ¡ 7x ¡ 3 = 0 1 3 Use technology to solve: a (x ¡ 2)(x + 1) = 3x ¡ 4 b 2x ¡ = 5 x 4 Using the discriminant only, determine the nature of the solutions of: b 3x2 ¡ 24x + 48 = 0 a 2x2 ¡ 5x ¡ 7 = 0

2 Solve the following equations: a

b

5 Find the values of m for which 2x2 ¡ 3x + m = 0 has: a a repeated root b two distinct real roots 6 If AB is the same length as CD, BC is 2 cm shorter than AB, and BE is 7 cm in length, find the length of AB.

c

no real roots D E

A

C

B

7 Find the length of the hypotenuse of a right angled triangle with one leg 7 cm longer than the other and the hypotenuse 2 cm longer than the longer leg. 8 Find the y-intercept of the line with slope ¡3 that is tangential to the parabola y = 2x2 ¡ 5x + 1. 9 ax2 + [3 ¡ a]x ¡ 4 = 0 has roots which are real and positive. What values can a have?

REVIEW SET 6D 1 Use the vertex, axis of symmetry and y-intercept to graph: b y = ¡ 12 (x + 4)2 + 6 a y = (x ¡ 2)2 ¡ 4 2 For the quadratic y = 2x2 + 4x ¡ 1, find: a the equation of the axis of symmetry b c the axes intercepts. d

the coordinates of the vertex Hence sketch the graph.

3 Use the discriminant only to find the relationship between the graph and the x-axis for: b y = ¡3x2 ¡ 7x + 4 a y = 2x2 + 3x ¡ 7 4 Determine if the quadratic functions are positive definite, negative definite or neither: b y = 3x2 + x + 11 a y = ¡2x2 + 3x + 2 5 Find the equation of the quadratic relation with graph:

a

y

-3

3

b

y

(2, 25)

x 1

-27

x

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1 6 The sum of a number and its reciprocal is 2 30 . Find the number.

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7 An open square container is made by cutting 4 cm square pieces out of a piece of tinplate. If the capacity is 120 cm3 , find the size of the original piece of tinplate. 8 Find the points where y = ¡x2 ¡ 5x + 3 and y = x2 + 3x + 11 meet. 9 Show that no line with a y-intercept of (0, 10) will ever be tangential to the curve with equation y = 3x2 + 7x ¡ 2. 1 1 10 Find all quadratic equations which have roots m ¡ and n ¡ given that m n m 2 and n are the roots of 3x ¡ 2x ¡ 2 = 0.

REVIEW SET 6E 1 Find the equation of the quadratic relation with graph:

a

b

y

y

18

5 x

-3

c

1

(2,-20)

x

d

y

y

7

2

x

-2

-3

x¡=¡4

x

2 Find an expression for a quadratic which cuts the x-axis at 3 and ¡2 and has y-intercept 24. Give your answer in the form y = ax2 + bx + c. 3 Find, in the form y = ax2 + bx + c, the equation of the quadratic whose graph: a touches the x-axis at 4 and passes through (2, 12). b has vertex (¡4, 1) and passes through (1, 11). 4 Find the maximum or minimum value of the following quadratics, and the correspondb y = ¡2x2 ¡ 5x + 2 ing value of x: a y = 3x2 + 4x + 7 5 For what values of k would the graph of y = x2 ¡ 2x + k cut the x-axis twice? Check your answer(s) using technology. ym

6 600 m of fencing are used to construct xm 6 rectangular animal pens as shown. 600 ¡ 8x a Show that y = . 9 b Find the area A of each pen in terms of x. c Find the dimensions of each pen if each pen is to have maximum area. d What is the maximum area of each pen? 7 Show that the lines with equations y = ¡5x + k y = x2 ¡ 3x + c if and only if c ¡ k = 1.

are tangents to the parabola

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8 4x2 ¡ 3x ¡ 3 = 0 has roots p, q. Find all quadratic equations with roots p3 and q 3 .

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Chapter

7

Complex numbers and polynomials

A Solutions of real quadratics with ¢ < 0 B Complex numbers C Real polynomials D Roots, zeros and factors E Graphing polynomials F Theorems for real polynomials

Contents:

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COMPLEX NUMBERS AND POLYNOMIALS (Chapter 7)

A

SOLUTIONS OF REAL QUADRATICS WITH ¢ < 0

In Chapter 6, we determined that: If ax2 + bx + c = 0, a 6= 0 and a, b, c 2 R , then the solutions or roots are found p ¡b § ¢ using the formula x = where ¢ = b2 ¡ 4ac is known as the discriminant. 2a ² ¢>0 ² ¢=0 ² ¢ 0. 7 For z = 2 ¡ i and w = 1 + 3i, find in the form a + bi where a and b are real: z i w a b c d z ¡2 w z iz 8 Simplify:

i 1 ¡ 2i

a

i(2 ¡ i) 3 ¡ 2i

b

9 If z = 2 + i and w = ¡1 + 2i, find: a Im(4z ¡ 3w) b Re(zw)

c

Im(iz 2 )

c

1 2 ¡ 2¡i 2+i d

Re

³z´ w

10 Check your answers to questions 1 to 4 and 7 to 9 using technology.

EQUALITY OF COMPLEX NUMBERS Two complex numbers are equal when their real parts are equal and their imaginary parts are equal. a + bi = c + di , a = c and b = d.

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Suppose b 6= d. Now if a + bi = c + di where a, b, c and d are real, then bi ¡ di = c ¡ a ) i(b ¡ d) = c ¡ a c¡a fas b 6= dg ) i= b¡d and this is false as the RHS is real and the LHS is imaginary. Thus, the supposition is false and hence b = d and furthermore a = c.

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Proof:

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Example 8 If (x + yi)(2 ¡ i) = ¡i and x, y are real, determine the values of x and y. ¡i 2¡i µ ¶µ ¶ ¡i 2+i i.e., x + yi = 2¡i 2+i

If (x + yi)(2 ¡ i) = ¡i, then x + yi =

¡2i ¡ i2 1 ¡ 2i = 4+1 5 1 2 x + yi = 5 ¡ 5 i

)

x + yi =

)

and so x = 15 , y = ¡ 25 .

Example 9 Find real numbers x and y for which (x + 2i)(1 ¡ i) = 5 + yi (x + 2i)(1 ¡ i) = 5 + yi x ¡ xi + 2i + 2 = 5 + yi ) [x + 2] + [2 ¡ x]i = 5 + yi x + 2 = 5 and 2 ¡ x = y fequating real and imaginary partsg ) x = 3 and y = ¡1 )

)

EXERCISE 7B.2 1 Find real numbers x and y such that: a 2x + 3yi = ¡x ¡ 6i c (x + yi)(2 ¡ i) = 8 + i

b d

2 Find x and y if x, y 2 R and: a 2(x + yi) = x ¡ yi c (x + i)(3 ¡ iy) = 1 + 13i

b d

x2 + xi = 4 ¡ 2i (3 + 2i)(x + yi) = ¡i

(x + 2i)(y ¡ i) = ¡4 ¡ 7i (x + yi)(2 + i) = 2x ¡ (y + 1)i p 3 Write z in the form a + bi where a, b 2 R and i = ¡1, if the complex number z satisfies the equation 3z + 17i = iz + 11 p 4 z= 4 The complex number z is a solution of the equation + 7 ¡ 2i: 1+i Express z in the form a + bi where a and b 2 Z . 5 Find the real values of m and n for which 3(m + ni) = n ¡ 2mi ¡ (1 ¡ 2i).

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3i 6 Express z = p + 1 in the form a + bi where a, b 2 R , giving the exact 2¡i values of the real and imaginary parts of z.

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COMPLEX CONJUGATES Complex numbers a + bi and a ¡ bi are called complex conjugates. If z = a + bi we write its conjugate as z ¤ = a ¡ bi. Recall from page 181 that the conjugate is important for division: zw¤ z z w¤ = which makes the denominator real. = w w w¤ ww¤ Complex conjugates appear as the solutions of real quadratic equations of the form ax2 + bx + c = 0 where the discriminant ¢ = b2 ¡ 4ac is negative. For example: ² ²

x2 ¡ 2x + 5 = 0

has and has and

x2 + 4 = 0

¢ = (¡2)2 ¡ 4(1)(5) = ¡16 the solutions are x = 1 + 2i and 1 ¡ 2i ¢ = 02 ¡ 4(1)(4) = ¡16 the solutions are x = 2i and ¡2i

² Quadratics with real coefficients are called real quadratics. This does not necessarily mean that its zeros are real. ² If a quadratic equation has rational p coefficients and an irrational root of p the form c + d n, then c ¡ d n is also a root. These roots are radical conjugates. ² If a real quadratic equation has ¢ < 0 and c + di is a complex root then c ¡ di is also a root. These roots are complex conjugates.

Note:

If c + di and c ¡ di are roots of a quadratic equation, then the quadratic equation is a(x2 ¡ 2cx + (c2 + d2 )) = 0 for some constant a 6= 0.

Theorem:

Proof: The sum of the roots = 2c and the product = (c + di)(c ¡ di) = c2 + d2 ) x2 ¡ (sum) x + (product) = 0 ) x2 ¡ 2cx + (c2 + d2 ) = 0 Alternatively: If c + di and c ¡ di are roots then (x ¡ [c + di])(x ¡ [c ¡ di]) = 0 ) (x ¡ c ¡ di)(x ¡ c + di) = 0 ) (x ¡ c)2 ¡ d2 i2 = 0 ) x2 ¡ 2cx + c2 + d2 = 0 In general, a(x2 ¡ 2cx + c2 + d2 ) = 0 for some constant a 6= 0.

Example 10 Find all quadratic equations with real coefficients having 1 ¡ 2i as a root. As 1 ¡ 2i is a root, 1 + 2i is also a root. Sum of roots = 1 ¡ 2i + 1 + 2i Product of roots = (1 ¡ 2i)(1 + 2i) =2 =1+4 2 =5 So, as x ¡ (sum) x + (product) = 0,

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a(x2 ¡ 2x + 5) = 0, a 6= 0 gives all possible equations.

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Note:

The sum of complex conjugates c + di and c ¡ di is 2c which is real. The product is (c + di)(c ¡ di) = c2 + d2 which is also real.

Example 11 Find exact values of a and b if

p 2 + i is a root of x2 + ax + b = 0, a, b 2 R .

Since a and b are real, the quadratic has real coefficients p 2 ¡ i is also a root ) p p p ) sum of roots = 2 + i + 2 ¡ i = 2 2 p p product of roots = ( 2 + i)( 2 ¡ i) = 2 + 1 = 3 p Thus a = ¡2 2 and b = 3.

EXERCISE 7B.3 1 Find all quadratic equations with real coefficients and roots of: a 3§i b 1 § 3i c ¡2 § 5i p p 2 e 2§ 3 f 0 and ¡ 3 g §i 2

d

p 2§i

h

¡6 § i

2 Find exact values of a and b if: a 3 + i is a root of x2 + ax + b = 0, where a and b are real p b 1 ¡ 2 is a root of x2 + ax + b = 0, where a and b are rational c a + ai is a root of x2 + 4x + b = 0 where a and b are real. [Careful!]

INVESTIGATION 1

PROPERTIES OF CONJUGATES

The purpose of this investigation is to discover any properties that complex conjugates might have. What to do: 1 Given z1 = 1 ¡ i and z2 = 2 + i find: a z1¤ b z2¤ c (z1¤ )¤ d (z2¤ )¤ g (z1 ¡ z2 )¤ ¡ ¢¤ m z12

h z1¤ ¡ z2¤

i (z1 z2 )¤ ¡ ¢¤ o z23

n (z1¤ )2

j z1¤ z2¤ p (z2¤ )3

e (z1 +z2 )¤ µ ¶¤ z1 k z2

f z1¤ + z2¤ z¤ l 1¤ z2

2 Repeat 1 with z1 and z2 of your choice. 3 From 1 and 2 formulate possible rules of conjugates.

PROPERTIES OF CONJUGATES From Investigation 1 you should have found the following rules for complex conjugates:

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(z1 + z2 ) ¤ = z1¤ + z2¤ and (z1 ¡ z2 ) ¤ = z1¤ ¡ z2¤ µ ¶¤ z1 z¤ ¤ ¤ ¤ (z1 z2 ) = z1 £ z2 and = 1¤ , z2 6= 0 z2 z2

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(z n ) ¤ = (z ¤ )n

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z +z¤

for integers n = 1, 2 and 3

and zz ¤

are real.

Example 12 Show that (z1 + z2 ) ¤ = z1¤ + z2¤

for all complex numbers z1 and z2 . z1¤ = a ¡ bi and z2¤ = c ¡ di

)

Let z1 = a + bi and z2 = c + di

(z1 + z2 ) ¤ = (a + c) ¡ (b + d)i = a + c ¡ bi ¡ di = a ¡ bi + c ¡ di = z1¤ + z2¤

)

Now z1 + z2 = (a + c) + (b + d)i

Example 13 Show that (z1 z2 ) ¤ = z1¤ £ z2¤ for all complex numbers z1 and z2 .

Let z1 = a + bi and z2 = c + di ) z1 z2 = (a + bi)(c + di) = ac + adi + bci + bdi2 = [ac ¡ bd] + i[ad + bc] Thus (z1 z2 ) ¤ = [ac ¡ bd] ¡ i[ad + bc] ...... (1) Now z1¤ £ z2¤ = (a ¡ bi)(c ¡ di) = ac ¡ adi ¡ bci + bdi2 = [ac ¡ bd] ¡ i[ad + bc] ...... (2) From (1) and (2), (z1 z2 ) ¤ = z1¤ £ z2¤

EXERCISE 7B.4 1 Show that (z1 ¡ z2 ) ¤ = z1¤ ¡ z2¤

for all complex numbers z1 and z2 .

2 Simplify the expression (w ¤ ¡ z) ¤ ¡ (w ¡ 2z ¤ ) using the properties of conjugates. 3 It is known that a complex number z satisfies the equation z ¤ = ¡z. Show that z is either purely imaginary or zero. 4 If z1 = a + bi and z2 = c + di: µ ¶¤ z1 z¤ z1 (in form X + Y i) b show that = 1¤ for all z1 and z2 6= 0. a find z2 z2 z2 µ ¶¤ µ ¶¤ z1 z1 z1¤ = ¤ is to start with £ z2¤ . 5 An easier way of proving z2 z2 z2 Show how this can be done, remembering we have already proved that “the conjugate of a product is the product of the conjugates” in Example 13.

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6 Prove that for all complex numbers z and w: b zw ¤ ¡ z ¤ w a zw ¤ + z ¤ w is always real

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a If z = a + bi find z 2 in the form X + Y i. ¡ ¢¤ b Hence, show that z 2 = (z ¤ )2 for all complex numbers z. c Repeat a and b but for z 3 instead of z 2 .

7

8 w= a

z¡1 where z = a + bi. Find the conditions under which: z¤ +1 w is real b w is purely imaginary.

CONJUGATE GENERALISATIONS Notice that (z1 + z2 + z3 ) ¤ = (z1 + z2 ) ¤ + z3¤ ftreating z1 + z2 as one complex numberg ...... (1) = z1¤ + z2¤ + z3¤ Likewise (z1 + z2 + z3 + z4 ) ¤ = (z1 + z2 + z3 ) ¤ + z4¤ = z1¤ + z2¤ + z3¤ + z4¤

ffrom (1)g

Since there is no reason why this process cannot continue for the conjugate of 5, 6, 7, .... complex numbers we generalise to: (z1 + z2 + z3 + :::::: + zn ) ¤ = z1¤ + z2¤ + z3¤ + :::::: + zn¤ : The process of obtaining the general case from observing the simpler cases when n = 1, 2, 3, 4, ..... is called mathematical induction. Proof by the Principle of Mathematical Induction is an exercise that could be undertaken after the completion of Chapter 9. This is a more formal treatment and constitutes a proper proof.

EXERCISE 7B.5 a b c d

1

Assuming Show that What is the What is the

(z1 z2 ) ¤ = z1¤ z2¤ , explain why (z1 z2 z3 ) ¤ = z1¤ z2¤ z3¤ . (z1 z2 z3 z4 ) ¤ = z1¤ z2¤ z3¤ z4¤ from a. inductive generalisation of your results in a and b? result of letting all zi values be equal to z in c?

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for all positive integers n

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(z1 + z2 + z3 + ::::: + zn ) ¤ = z1¤ + z2¤ + z3¤ + ::::: + zn¤ and (z1 z2 z3 ::::zn ) ¤ = z1¤ z2¤ z3¤ :::: zn¤

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If z is any complex number then z + z ¤ is real and zz ¤ is real. (z ¤ ) ¤ = z If z1 and z2 are any complex numbers then I (z1 ¡ z2 ) ¤ = z1¤ ¡ z2¤ I (z1 + z2 ) ¤ = z1¤ + z2¤ µ ¶¤ z1 z¤ ¤ ¤ ¤ I (z1 z2 ) = z1 z2 I = 1¤ z2 z2

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SUMMARY OF CONJUGATE DISCOVERIES

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C

REAL POLYNOMIALS

Up to this point we have studied linear and quadratic polynomial functions at some depth, with perhaps occasional reference to cubic and quartic polynomials. We now turn our attention to general polynomials with real coefficients. The degree of a polynomial is its highest power of the variable. Polynomials ax + b, a 6= 0 ax2 + bx + c, a 6= 0 ax3 + bx2 + cx + d, a 6= 0 ax4 + bx3 + cx2 + dx + e, a 6= 0

Degree 1 2 3 4

Name linear quadratic cubic quartic

a is the leading coefficient and the term not containing the variable x is the constant term. A real polynomial has all its coefficients as real numbers. (They do not p contain i where i = ¡1:)

OPERATIONS WITH POLYNOMIALS ADDITION AND SUBTRACTION

To add (or subtract) two polynomials we add (or subtract) ‘like terms’.

Example 14 If P (x) = x3 ¡ 2x2 + 3x ¡ 5 and Q(x) = 2x3 + x2 ¡ 11 find: a P (x) + Q(x) b P (x) ¡ Q(x) a

It is a good idea to place brackets around expressions which are subtracted.

P (x) + Q(x) b P (x) ¡ Q(x) 3 2 = x ¡ 2x + 3x ¡ 5 = x3 ¡ 2x2 + 3x ¡ 5 ¡ [2x3 + x2 ¡ 11] ¡ 11 + 2x3 + x2 = x3 ¡ 2x2 + 3x ¡ 5 ¡ 2x3 ¡ x2 + 11 = 3x3 ¡ x2 + 3x ¡ 16 = ¡x3 ¡ 3x2 + 3x + 6

SCALAR MULTIPLICATION To multiply a polynomial by a scalar (constant) we multiply each term by the scalar.

b

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¡2P (x) = ¡2(x4 ¡ 2x3 + 4x + 7) = ¡2x4 + 4x3 ¡ 8x ¡ 14

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3P (x) = 3(x4 ¡ 2x3 + 4x + 7) = 3x4 ¡ 6x3 + 12x + 21

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a 3P (x)

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If P (x) = x4 ¡ 2x3 + 4x + 7 find:

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Example 15

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POLYNOMIAL MULTIPLICATION

To multiply two polynomials, we multiply every term of the first polynomial by every term of the second polynomial and then collect like terms.

Example 16 If P (x) = x3 ¡ 2x + 4 and Q(x) = 2x2 + 3x ¡ 5, find P (x) Q(x). P (x)Q(x) = (x3 ¡ 2x + 4)(2x2 + 3x ¡ 5) = x3 (2x2 + 3x ¡ 5) ¡ 2x(2x2 + 3x ¡ 5) + 4(2x2 + 3x ¡ 5) = 2x5 + 3x4 ¡ 5x3 ¡ 4x3 ¡ 6x2 + 10x + 8x2 + 12x ¡ 20 = 2x5 + 3x4 ¡ 9x3 + 2x2 + 22x ¡ 20

EXERCISE 7C.1 1 If P (x) = x2 + 2x + 3 and Q(x) = 4x2 + 5x + 6, find in simplest form: a 3P (x) b P (x) + Q(x) c P (x) ¡ 2Q(x) d P (x)Q(x) 2 If f(x) = x2 ¡ x + 2 and g(x) = x3 ¡ 3x + 5, find in simplest form: a f(x) + g(x) b g(x) ¡ f(x) c 2f (x) + 3g(x) d g(x) + xf(x) e f(x)g(x) f [f (x)]2 3 Expand and simplify: a (x2 ¡ 2x + 3)(2x + 1) c (x + 2)3 e (2x ¡ 1)4

b d f

(x ¡ 1)2 (x2 + 3x ¡ 2) (2x2 ¡ x + 3)2 (3x ¡ 2)2 (2x + 1)(x ¡ 4)

NOTE ON SYNTHETIC MULTIPLICATION (OPTIONAL)

Polynomial multiplication can be performed using the coefficients only. For example, for (x3 + 2x ¡ 5)(2x + 3) we detach coefficients and multiply. It is different from the ordinary multiplication of large numbers because we sometimes have negative coefficients and because we do not carry tens into the next column.

3 0 3

x4

x3

x2

2 2 6 ¡10 ¡4 x

coefficients of x3 + 2x ¡ 5 coefficients of 2x + 3

¡5 3 ¡15

So (x3 + 2x ¡ 5)(2x + 3) = 2x4 + 3x3 + 4x2 ¡ 4x ¡ 15.

¡15 constants

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4 Find the following products: a (2x2 ¡ 3x + 5)(3x ¡ 1) c (2x2 + 3x + 2)(5 ¡ x)

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(4x2 ¡ x + 2)(2x + 5) (x ¡ 2)2 (2x + 1)

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e g

(x2 ¡ 3x + 2)(2x2 + 4x ¡ 1) (x2 ¡ x + 3)2 h (2x2 + x ¡ 4)2

f i

(3x2 ¡ x + 2)(5x2 + 2x ¡ 3) (2x + 5)3 j (x3 + x2 ¡ 2)2

DIVISION OF POLYNOMIALS The division of polynomials is a more difficult process. We can divide a polynomial by another polynomial using an algorithm similar to that used for division of whole numbers. The division process is only sensible if we divide a polynomial of degree n by another of degree n or less. DIVISION BY LINEARS

Consider (2x2 + 3x + 4)(x + 2) + 7. If we expand this expression we get (2x2 + 3x + 4)(x + 2) + 7 = 2x3 + 7x2 + 10x + 15. Now consider 2x3 + 7x2 + 10x + 15 divided by x + 2 (2x2 + 3x + 4)(x + 2) + 7 2x3 + 7x2 + 10x + 15 = x+2 x+2

i.e., )

2x3 + 7x2 + 10x + 15 (2x2 + 3x + 4)(x + 2) 7 = + x+2 x+2 x+2

)

7 2x3 + 7x2 + 10x + 15 2 = 2x | +{z3x + 4} + x + 2 x+2

remainder divisor

quotient

DIVISION ALGORITHM

Division may be performed directly using the following algorithm: 2x2 + x+2 2x3 + 7x2 + ¡ (2x3 + 4x2 ) 3x2 + ¡ (3x2 +

Step 1: What do we multiply x by to get 2x3 ? The answer is 2x2 , and 2x2 (x + 2) = 2x3 + 4x2 .

3x + 4 10x + 15

10x 6x) 4x + 15 ¡ (4x + 8) 7

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Step 3: Bring down the 10x to obtain 3x2 + 10x. Return to Step 1 with the question: “What must we multiply x by to get 3x2 ? ” The answer is 3x, and 3x(x + 2) = 3x2 + 6x etc.

constants

4 15

The result can also be achieved by leaving out the variable, as shown alongside.

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15 8) 7

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10 6) 4 ¡ (4

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2 ¡ (2

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x3 s

Step 2: Subtract 2x3 + 4x2 from 2x3 + 7x2 . The answer is 3x2 .

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Either way,

2x3 + 7x2 + 10x + 15 7 = 2x2 + 3x + 4 + , x+2 x+2 where x + 2 2x2 + 3x + 4 and 7

is called the divisor, is called the quotient, is called the remainder.

if P (x) is divided by ax + b until a constant remainder R is obtained,

In general,

P (x) R = Q(x) + ax + b ax + b

where ax + b Q(x) R

is the divisor, is the quotient, and is the remainder.

Notice that P (x) = Q(x) £ (ax + b) + R:

Example 17 x3 ¡ x2 ¡ 3x ¡ 5 . x¡3

Find the quotient and remainder for x2 + 2x + 3 x¡3 x3 ¡ x2 ¡ 3x ¡ 5 ¡ (x3 ¡ 3x2 )

) quotient is x2 + 2x + 3 and remainder is 4.

2x2 ¡ 3x ¡ (2x2 ¡ 6x) 3x ¡ 5 ¡ (3x ¡ 9) 4

x3 ¡ x2 ¡ 3x ¡ 5 4 = x2 + 2x + 3 + x¡3 x¡3

Thus

So, x3 ¡ x2 ¡ 3x ¡ 5 = (x2 + 2x + 3)(x ¡ 3) + 4:

(Check by expanding and simplifying the RHS.)

Example 18 x4 + 2x2 ¡ 1 : x+3

x3 ¡ 3x2 + x+3 x4 + 0x3 + 2x2 + ¡ (x4 + 3x3 ) ¡ 3x3 + 2x2 ¡(¡ 3x3 ¡ 9x2 ) 11x2 + ¡ (11x2 +

Notice the insertion of 0x3 and 0x. Why?

11x ¡ 33 0x ¡ 1

= x3 ¡ 3x2 + 11x ¡ 33 +

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¡ 33x ¡ 1 ¡ (¡ 33x ¡ 99) 98

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0x 33x)

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Perform the division

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EXERCISE 7C.2 1 Find the quotient and remainder for: x2 + 2x ¡ 3 x+2

a

2 Perform the divisions: x2 ¡ 3x + 6 a x¡4 2x3 + 3x2 ¡ 3x ¡ 2 2x + 1

d

3 Perform the divisions: x2 + 5 a x¡2 x3 + 2x2 ¡ 5x + 2 x¡1

d

b

x2 ¡ 5x + 1 x¡1

c

2x3 + 6x2 ¡ 4x + 3 x¡2

b

x2 + 4x ¡ 11 x+3

c

2x2 ¡ 7x + 2 x¡2

e

3x3 + 11x2 + 8x + 7 3x ¡ 1

f

2x4 ¡ x3 ¡ x2 + 7x + 4 2x + 3

b

2x2 + 3x x+1

c

3x2 + 2x ¡ 5 x+2

e

2x3 ¡ x x+4

f

x3 + x2 ¡ 5 x¡2

DIVISION BY QUADRATICS As with division by linears we can use the division algorithm to divide polynomials by quadratics. The division process stops when the remainder has degree less than that of the divisor, P (x) ex + f = Q(x) + 2 i.e., 2 a + bx + c ax + bx + c The remainder will be linear if e 6= 0 or constant if e = 0.

Example 19 Find the quotient and remainder for

x4 + 4x3 ¡ x + 1 . x2 ¡ x + 1

x2 +5x +4 x2 ¡ x + 1 x4 +4x3 +0x2 ¡x +1 ¡(x4 ¡x3 + x2 ) 5x3 ¡ x2 ¡x ¡(5x3 ¡5x2 +5x) 4x2 ¡6x +1 ¡(4x2 ¡4x +4)

) quotient is x2 + 5x + 4 and remainder is ¡2x ¡ 3 So, x4 + 4x3 ¡ x + 1 = (x2 ¡ x + 1)(x2 + 5x + 4) ¡ 2x ¡ 3

¡2x ¡3

EXERCISE 7C.3 1 Find the quotient and remainder for:

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2 Carry out the following divisions and also write each in the form P (x) = D(x)Q(x) + R(x): a

x2 ¡ x + 1 x2 + x + 1

b

d

2x3 ¡ x + 6 (x ¡ 1)2

e

x3 +2

c

x4 + 3x2 + x ¡ 1 x2 ¡ x + 1

x4 (x + 1)2

f

x4 ¡ 2x3 + x + 5 (x ¡ 1)(x + 2)

x2

3 P (x) = (x ¡ 2)(x2 + 2x + 3) + 7. What is the quotient and remainder when P (x) is divided by x ¡ 2? 4 Given that f (x) = (x ¡ 1)(x + 2)(x2 ¡ 3x + 5) + 15 ¡ 10x, find the quotient and remainder when f (x) is divided by x2 + x ¡ 2. PRINTABLE SECTION

SYNTHETIC DIVISION (OPTIONAL)

Click on the icon for an exercise involving a synthetic division process for the division of a polynomial by a linear.

D

ROOTS, ZEROS AND FACTORS

ROOTS AND ZEROS A zero of a polynomial is a value of the variable which makes the polynomial equal to zero. The roots of a polynomial equation are values of the variable which satisfy or are solutions to the equation.

The roots of P (x) = 0 are the zeros of P (x) and the x-intercepts of the graph of y = P (x). If x = 2,

x3 + 2x2 ¡ 3x ¡ 10 = 8 + 8 ¡ 6 ¡ 10 =0

) 2 is a zero of x3 + 2x2 ¡ 3x ¡ 10 and 2 is a root of x3 + 2x2 ¡ 3x ¡ 10 = 0:

® is a zero of polynomial P (x) , ® is a root (or solution) of P (x) = 0

Note:

P (®) = 0. , P (®) = 0.

Example 20 a x2 ¡ 4x + 53

Find the zeros of: a

b z 3 + 3z b

We wish to find x such that x2 ¡ 4x + 53 = 0 p 4 § 16 ¡ 4(1)(53) ) x= 2 4 § 14i ) x= = 2 § 7i 2

We wish to find z such that z 3 + 3z = 0 ) z(z 2 + 3) = 0 p p ) z(z + i 3)(z ¡ i 3) = 0 p ) z = 0 or §i 3

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If P (x) = (x + 1)(2x ¡ 1)(x + 2), then (x + 1), (2x ¡ 1) and (x + 2) are its linear factors.

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Likewise P (x) = (x + 3)2 (2x + 3) has been factorised into 3 linear factors, one of which is repeated. In general, (x¡®) is a factor of polynomial P (x) , there exists a polynomial Q(x) such that P (x) = (x ¡ ®)Q(x):

Example 21 a z 2 + 4z + 9

Factorise: a

b 2z 3 + 5z 2 ¡ 3z ?

z 2 + 4z + 9 is zero when b p p ¡4 § 16 ¡ 4(1)(9) = ¡2 § i 5 ) z= 2 p p 2 ) z + 4z + 9 = (z ¡ [¡2 + i 5])(z ¡ [¡2 ¡ i 5]) p p = (z + 2 ¡ i 5)(z + 2 + i 5)

Example 22

Find all cubic polynomials with zeros

The zeros ¡3 § 2i have

1 2,

2z 3 + 5z 2 ¡ 3z = z(2z 2 + 5z ¡ 3) = z(2z ¡ 1)(z + 3)

¡3 § 2i:

sum = ¡3 + 2i ¡ 3 ¡ 2i = ¡6 and product = (¡3 + 2i)(¡3 ¡ 2i) = 13

and ) come from the quadratic factor z 2 + 6z + 13 1 2 comes from the linear factor 2z ¡ 1 )

P (z) = a(2z ¡ 1)(z 2 + 6z + 13), a 6= 0.

Example 23

Find all quartic polynomials with zeros of 2, ¡ 13 , ¡1 §

The zeros ¡1 §

p 5.

p p 5 ¡ 1 ¡ 5 = ¡2 and p p product = (¡1 + 5)(¡1 ¡ 5) = ¡4

p 5 have

sum = ¡1 +

and ) come from the quadratic factor z 2 + 2z ¡ 4 The zeros 2 and ¡ 13 come from the linear factors (z ¡ 2) and (3z + 1) )

P (z) = a(z ¡ 2)(3z + 1)(z 2 + 2z ¡ 4), a 6= 0.

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¡2z(z 2 ¡ 2z + 2) = 0 z 4 = 3z 2 + 10

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(2x + 1)(x2 + 3) = 0 z 3 + 5z = 0

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2 Find the roots of: a 5x2 = 3x + 2 d x3 = 5x

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z 2 ¡ 6z + 6 z 4 + 4z 2 ¡ 5

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x2 + 6x + 10 z 3 + 2z

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1 Find the zeros of: a 2x2 ¡ 5x ¡ 12 d x3 ¡ 4x

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EXERCISE 7D.1

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3 Find the linear factors of: a 2x2 ¡ 7x ¡ 15 d 6z 3 ¡ z 2 ¡ 2z

b e

z 2 ¡ 6z + 16 z 4 ¡ 6z 2 + 5

c f

x3 + 2x2 ¡ 4x z4 ¡ z2 ¡ 2

4 If P (x) = a(x ¡ ®)(x ¡ ¯)(x ¡ °) then ®, ¯ and ° are its zeros. Check that the above statement is correct by finding P (®), P (¯) and P (°). 5 Find all cubic polynomials with zeros of: a §2, 3 b ¡2, §i

c

6 Find all quartic polynomials with zeros of: p p b 2, ¡1, §i 3 c a §1, § 2

3, ¡1 § i

d

¡1, ¡2 §

p § 3, 1 § i

d

2§

p 2

p 5, ¡2 § 3i

POLYNOMIAL EQUALITY Polynomials are equal if and only if they generate the same y-value for each x-value. This means that graphs of equal polynomials should be identical. Two polynomials are equal if and only if they have the same degree (order) and corresponding terms have equal coefficients. For example, if 2x3 + 3x2 ¡ 4x + 6 = ax3 + bx2 + cx + d, then a = 2, b = 3, c = ¡4 and d = 6. EQUATING COEFFICIENTS

If we know that two polynomials are equal then we can ‘equate coefficients’ in order to find unknown coefficients.

Example 24 Find constants a, b and c given that: 6x3 + 7x2 ¡ 19x + 7 = (2x ¡ 1)(ax2 + bx + c) for all x. If 6x3 + 7x2 ¡ 19x + 7 = (2x ¡ 1)(ax2 + bx + c) then 6x3 + 7x2 ¡ 19x + 7 = 2ax3 + 2bx2 + 2cx ¡ ax2 ¡ bx ¡ c ) 6x3 + 7x2 ¡ 19x + 7 = 2ax3 + [2b ¡ a]x2 + [2c ¡ b]x ¡ c Since this is true for all x, we equate coefficients ) 2a = 6}, 2b a = 7}, 2c | {z | ¡ b{z= ¡19} and |7 ={z¡c} | ¡{z x2 s xs constants x3 s

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a = 3 and c = ¡7 and consequently 2b | ¡ 3 = 7 and {z¡ 14 ¡ b = ¡19} b=5 in both equations So, a = 3, b = 5 and c = ¡7.

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Example 25 Find a and b if z 4 + 9 = (z 2 + az + 3)(z 2 + bz + 3) for all z. z 4 + 9 = (z 2 + az + 3)(z 2 + bz + 3) for all z z 4 + 9 = z 4 + bz 3 + 3z 2 +az 3 + abz 2 + 3az +3z 2 + 3bz + 9

)

) z 4 + 9 = z 4 + [a + b]z 3 + [ab + 6]z 2 + [3a + 3b]z + 9 8 a + b = 0 ...... (1) < ab + 6 = 0 ...... (2) Equating coefficients gives : 3a + 3b = 0 ...... (3)

for all z

When solving more equations than unknowns simultaneously, we must check that any solutions fit all equations.¡ If they do not, there are no solutions.

fz 3 sg fz 2 sg fz sg

From (1) and (3) we see that b = ¡a So, in (2) a(¡a) + 6 = 0 ) a2 = 6 p p ) a = § 6 and so b = ¨ 6 p p p p ) a = 6, b = ¡ 6 or a = ¡ 6, b = 6

Example 26 x + 3 is a factor of P (x) = x3 + ax2 ¡ 7x + 6. Find a and the other factors. Since x + 3 is a factor, x3 + ax2 ¡ 7x + 6 = (x + 3)(x2 + bx + 2) x3

for some constant b

6

= x3 + bx2 + 2x + 3x2 + 3bx + 6 = x3 + [b + 3]x2 + [3b + 2]x + 6 3b + 2 = ¡7 and a = b + 3 ) b = ¡3 and ) a = 0

Equating coefficients gives

P (x) = (x + 3)(x2 ¡ 3x + 2) = (x + 3)(x ¡ 1)(x ¡ 2)

)

Example 27 2x + 3 and x ¡ 1 are factors of 2x4 + ax3 ¡ 3x2 + bx + 3. Find a and b and all zeros of the polynomial. Since 2x + 3 and x ¡ 1 are factors, 2x4 + ax3 ¡ 3x2 + bx + 3 = (2x + 3)(x ¡ 1)(a quadratic) = (2x2 + x ¡ 3)(x2 + cx ¡ 1) for some c

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Equating coefficients of x2 gives:

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Equating coefficients of x3 : Equating coefficients of x:

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a = 2c + 1 = 4 + 1 = 5 b = ¡1 ¡ 3c ) b = ¡1 ¡ 6 = ¡7

) P (x) = (2x + 3)(x ¡ 1)(x2 + 2x ¡ 1) which has zeros of: p p p ¡2 § 4 ¡ 4(1)(¡1) ¡2 § 2 2 3 ¡ 2 , 1 and = = ¡1 § 2 2 2 p 3 ) the zeros are ¡ 2 , 1 and ¡1 § 2.

EXERCISE 7D.2 1 Find constants a, b and c given that: a 2x2 + 4x + 5 = ax2 + [2b ¡ 6]x + c for all x b 2x3 ¡ x2 + 6 = (x ¡ 1)2 (2x + a) + bx + c for all x. 2 Find a and b if: a z 4 + 4 = (z 2 + az + 2)(z 2 + bz + 2) for all z b 2z 4 + 5z 3 + 4z 2 + 7z + 6 = (z 2 + az + 2)(2z 2 + bz + 3) for all z 3 Show that z 4 + 64 can be factorised into two real quadratic factors of the form z 2 + az + 8 and z 2 + bz + 8, but cannot be factorised into two real quadratic factors of the form z 2 + az + 16 and z 2 + bz + 4. 4 Find real numbers a and b such that x4 ¡ 4x2 + 8x ¡ 4 = (x2 + ax + 2)(x2 + bx ¡ 2), and hence solve the equation x4 + 8x = 4x2 + 4. 5

a 2z ¡ 3 is a factor of 2z 3 ¡ z 2 + az ¡ 3. Find a and all zeros of the cubic. b 3z + 2 is a factor of 3z 3 ¡ z 2 + [a + 1]z + a. Find a and all the zeros of the cubic.

6

a Both 2x + 1 and x ¡ 2 are factors of P (x) = 2x4 + ax3 + bx2 ¡ 12x ¡ 8. Find a and b and all zeros of P (x). b x + 3 and 2x ¡ 1 are factors of 2x4 + ax3 + bx2 + ax + 3. Find a and b and hence determine all zeros of the quartic.

7

a x3 + 3x2 ¡ 9x + c has two identical linear factors. Prove that c is either 5 or ¡27 and factorise the cubic into linear factors in each case. b 3x3 + 4x2 ¡ x + m has two identical linear factors. Find m and find the zeros of the polynomial in all possible cases.

THE REMAINDER THEOREM Consider the cubic polynomial P (x) = x3 + 5x2 ¡ 11x + 3. If we divide P (x) by x ¡ 2, we find that: x3 + 5x2 ¡ 11x + 3 9 = x2 + 7x + 3 + x¡2 x¡2

remainder

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so on division by x ¡ 2, the remainder is 9:

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P (2) = 8 + 20 ¡ 22 + 3 = 9, which is the remainder.

Notice also that

By considering other examples like the one above we formulate the Remainder theorem.

THE REMAINDER THEOREM When a polynomial P (x) is divided by x ¡ k until a constant remainder R is obtained then R = P (k). P (x) = Q(x)(x ¡ k) + R P (k) = Q(k) £ 0 + R P (k) = R

By the division algorithm, Now, letting x = k, )

Proof:

Example 28 Use the Remainder theorem to find the remainder when x4 ¡ 3x3 + x ¡ 4 is divided by x + 2. If P (x) = x4 ¡ 3x3 + x ¡ 4, then P (¡2) = (¡2)4 ¡ 3(¡2)3 + (¡2) ¡ 4 = 16 + 24 ¡ 2 ¡ 4 = 34 ) when P (x) is divided by x + 2, the remainder is 34. fRemainder theoremg

Example 29 When P (x) is divided by x2 ¡ 3x + 7 the quotient is x2 + x ¡ 1 and the remainder R(x) is unknown. However, when P (x) is divided by x ¡ 2 the remainder is 29 and when divided by x + 1 the remainder is ¡16. Find R(x) in the form ax + b. As the divisor is x2 ¡ 3x + 7, +b P (x) = (x2 + x ¡ 1) (x2 ¡ 3x + 7) + ax{z } }| } | | {z {z Q(x) D(x) R(x) But P (2) = 29 and P (¡1) = ¡16 fRemainder theoremg ) (22 + 2 ¡ 1)(22 ¡ 6 + 7) + 2a + b = 29 2 and ((¡1) + (¡1) ¡ 1)((¡1)2 ¡ 3(¡1) + 7) + (¡a + b) = ¡16 )

(5)(5) + 2a + b = 29 (¡1)(11) ¡ a + b = ¡16 )

2a + b = 4 ¡a + b = ¡5

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Solving these gives a = 3 and b = ¡2, so R(x) = 3x ¡ 2.

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When using the Remainder theorem, it is important to realise that ² P (x) = (x + 2)Q(x) + 3 ² P (¡2) = 3 ² ‘P (x) divided by x + 2 leaves a remainder of 3’ are all equivalent statements.

EXERCISE 7D.3 1 Write two equivalent statements for: a If P (2) = 7, then ...... b If P (x) = (x + 3)Q(x) ¡ 8, then ...... c If P (x) divided by x ¡ 5 has a remainder of 11 then ...... 2 Without performing division, find the remainder when: a x3 + 2x2 ¡ 7x + 5 is divided by x ¡ 1 b x4 ¡ 2x2 + 3x ¡ 1 is divided by x + 2. 3 Find a given that: a when x3 ¡ 2x + a is divided by x ¡ 2, the remainder is 7 b when 2x3 + x2 + ax ¡ 5 is divided by x + 1, the remainder is ¡8. 4 Find a and b given that when x3 + 2x2 + ax + b is divided by x ¡ 1 the remainder is 4, and when divided by x + 2 the remainder is 16. 5 2xn +ax2 ¡6 leaves a remainder of ¡7 when divided by x¡1, and 129 when divided by x + 3. Find a and n given that n 2 Z + . 6 When P (z) is divided by z 2 ¡ 3z + 2 the remainder is 4z ¡ 7. Find the remainder when P (z) is divided by: a z¡1 b z ¡ 2. 7 When P (z) is divided by z + 1 the remainder is ¡8 and when divided by z ¡ 3 the remainder is 4. Find the remainder when P (z) is divided by (z ¡ 3)(z + 1). ¶ µ P (b) ¡ P (a) 8 If P (x) is divided by (x ¡ a)(x ¡ b), £ (x ¡ a) + P (a). prove that the remainder is: b¡a

THE FACTOR THEOREM ,

k is a zero of P (x)

(x ¡ k) is a factor of P (x).

Proof: k is a zero of P (x) , P (k) = 0 ,R=0 , P (x) = Q(x)(x ¡ k) , (x ¡ k) is a factor of P (x)

fdefinition of a zerog fRemainder theoremg fdivision algorithmg fdefinition of factorg

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The Factor theorem says that if 2 is a zero of P (x) then (x ¡ 2) is a factor of P (x) and vice versa.

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Example 30 Find k given that x ¡ 2 is a factor of x3 + kx2 ¡ 3x + 6. Hence, fully factorise x3 + kx2 ¡ 3x + 6. Let P (x) = x3 + kx2 ¡ 3x + 6 By the Factor theorem, as x ¡ 2 is a factor then P (2) = 0 ) (2)3 + k(2)2 ¡ 3(2) + 6 = 0 ) 8 + 4k = 0 and so k = ¡2 Now x3 ¡ 2x2 ¡ 3x + 6 = (x ¡ 2)(x2 + ax ¡ 3) for some constant a. Equating coefficients of x2 gives: Equating coefficients of x gives:

¡2 = ¡2 + a ¡3 = ¡2a ¡ 3

x3 ¡ 2x2 ¡ 3x + 6 = (x ¡ 2)(x2 ¡ 3) p p = (x ¡ 2)(x + 3)(x ¡ 3)

)

or Using synthetic division

)

i.e., a = 0 i.e., a = 0

2

1 0

k 2

¡3 2k + 4

6 4k + 2

1

k+2

2k + 1

4k + 8

P (2) = 4k + 8 and since P (2) = 0, k = ¡2 Now P (x) = (x ¡ 2)(x2 + [k + 2]x + [2k + 1]) = (x ¡ 2)(x2 ¡ 3) p p = (x ¡ 2)(x + 3)(x ¡ 3)

EXERCISE 7D.4 1 Find k and hence factorise the polynomial if: a 2x3 + x2 + kx ¡ 4 has a factor of x + 2 b x4 ¡ 3x3 ¡ kx2 + 6x has a factor of x ¡ 3. 2 Find a and b given that 2x3 + ax2 + bx + 5 has factors of x ¡ 1 and x + 5. 3

a 3 is a zero of P (z) = z 3 ¡ z 2 + [k ¡ 5]z + [k2 ¡ 7]. Find k and hence find all zeros of P (z). b Show that z ¡ 2 is a factor of P (z) = z 3 + mz2 + (3m ¡ 2)z ¡ 10m ¡ 4 for all values of m. For what values of m is (z ¡ 2)2 a factor of P (z)?

4

a Consider P (x) = x3 ¡ a3 where a is real. i Find P (a). What is the significance of this result? ii Factorise x3 ¡ a3 as the product of a real linear and a quadratic factor. b Now consider P (x) = x3 + a3 , where a is real. i Find P (¡a). What is the significance of this result? ii Factorise x3 + a3 as the product of a real linear and a quadratic factor.

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a Prove that “x + 1 is a factor of xn + 1 , n is odd.” b Find the real number a such that x ¡ 1 ¡ a is a factor of P (x) = x3 ¡ 3ax ¡ 9.

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E

GRAPHING POLYNOMIALS

In this section we are obviously only concerned with graphing real polynomials. Remember that these are polynomials for which all coefficients are real. Use of a graphics calculator or the graphing package provided will help in this section.

INVESTIGATION 2

CUBIC GRAPHS

Every cubic polynomial can be categorised into one of four types: Type Type Type Type

1: 2: 3: 4:

Three real, distinct zeros: P (x) = a(x ¡ ®)(x ¡ ¯)(x ¡ °), a 6= 0 6 0 Two real zeros, one repeated: P (x) = a(x ¡ ®)2 (x ¡ ¯), a = 3 One real zero repeated three times: P (x) = a(x ¡ ®) , a 6= 0 One real and two imaginary zeros: P (x) = (x ¡ ®)(ax2 + bx + c), ¢ = b2 ¡ 4ac < 0, a 6= 0

What to do:

1 Experiment with the graphs of Type 1 cubics. Clearly state the effect of changing both the size and sign of a. What is the geometrical significance of ®, ¯, and °? 2 Experiment with the graphs of Type 2 cubics. What is the geometrical significance of the squared factor? 3 Experiment with the graphs of Type 3 cubics. Do not forget to GRAPHING PACKAGE consider a¡>¡0 and a¡ 0, the graph has shape

or

. If a < 0 it is

or

.

² All cubics are continuous smooth curves. ² Every cubic polynomial must cut the x-axis at least once, and so has at least one real zero. ² For a cubic of the form P (x) = a(x ¡ ®)(x ¡ ¯)(x ¡ °) the graph has three distinct x-intercepts corresponding to the three real, distinct zeros ®, ¯ and °. The graph crosses over or cuts the x-axis at these points, as shown opposite. ² For a cubic of the form P (x) = a(x ¡ ®)2 (x ¡ ¯) the graph touches the x-axis at the repeated zero ® and cuts it at the other x-intercept ¯, as shown opposite.

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² For a cubic of the form P (x) = a(x ¡ ®)3 , the graph has only one x-intercept, ®. The graph is horizontal at this point, and the x-axis is a tangent to the curve, even though the curve crosses over it.

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² For a cubic of the form P (x) = (x ¡ ®)(ax2 + bx + c) where ¢ < 0, there is only one x-intercept, ®. The graph cuts the x-axis at this point. The other two zeros are imaginary and so do not show up on the graph.

x

Example 31 y

y

a

Find the equation of the cubic with graph:

b 2

4 x

-1

6 -3

x

We_

-8

The x-intercepts are ¡1, 2, 4 ) y = a(x + 1)(x ¡ 2)(x ¡ 4) But when x = 0, y = ¡8 ) a(1)(¡2)(¡4) = ¡8 ) a = ¡1

a

Touching at 23 indicates a squared factor (3x ¡ 2)2 . Other x-intercept is ¡3, so y = a(3x ¡ 2)2 (x + 3)

b

But when x = 0, y = 6 So, a(¡2)2 (3) = 6 and ) a =

So, y = ¡(x + 1)(x ¡ 2)(x ¡ 4) ²

Note:

So, y = 12 (3x ¡ 2)2 (x + 3)

If an x-intercept is not given, use P (x) = (x ¡ k)2 (ax + b) : | {z } most general form of a linear

x

2

Using P (x) = a(x ¡ k) (x + b) is more complicated. ²

1 2

?

If there is clearly only one x-intercept and that is given, use P (x) = (x ¡ k) (ax2 + bx + c) : | {z }

k

x k

most general form of a quadratic

What can you say about this quadratic?

Either graph is possible.

Example 32 Find the equation of the cubic which cuts the x-axis at 2, ¡3 and ¡4 and passes through the point (1, ¡40).

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The zeros are 2, ¡3 and ¡4, so y = a(x ¡ 2)(x + 3)(x + 4), a 6= 0. But when x = 1, y = ¡40 ) a(¡1)(4)(5) = ¡40 ) ¡20a = ¡40 ) a=2 So, the equation is y = 2(x ¡ 2)(x + 3)(x + 4):

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EXERCISE 7E.1 1 What is the geometrical significance of: a a single factor in P (x), such as (x ¡ ®) b a squared factor in P (x), such as (x ¡ ®)2 c a cubed factor in P (x), such as (x ¡ ®)3 ? 2 Find the equation of the cubic with graph: a b y

c

y

y -4

6

12

\Qw_

-\Qw_

3

x

x

-3 x

-1 2

3 -12

d

e

y

f

y

y -3

9 -5

5 -2

x

-2 -\Qw_

x

-12

-5

x

3

-4

3 Find the equation of the cubic whose graph: a cuts the x-axis at 3, 1, ¡2 and passes through (2, ¡4) b cuts the x-axis at ¡2, 0 and 12 and passes through (¡3, ¡21) c touches the x-axis at 1, cuts the x-axis at ¡2 and passes through (4, 54) d touches the x-axis at ¡ 23 , cuts the x-axis at 4 and passes through (¡1, ¡5). 4 Match the given graphs to the corresponding cubic function: a y = 2(x ¡ 1)(x + 2)(x + 4) b y = ¡(x + 1)(x ¡ 2)(x ¡ 4) c y = (x ¡ 1)(x ¡ 2)(x + 4) d y = ¡2(x ¡ 1)(x + 2)(x + 4) e y = ¡(x ¡ 1)(x + 2)(x + 4) f y = 2(x ¡ 1)(x ¡ 2)(x + 4) A B C y y y x

2 8 -4

-1

x 1

16

x -8

2 -4

D

1

E

y

2

F

y

y

16

8 x

-2 -4

4

x

-2

1

-4

-2

1 x

-4

1

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5 Find the equation of a real cubic polynomial which cuts: a the x-axis at 12 and ¡3, cuts the y-axis at 30 and passes through (1, ¡20) b the x-axis at 1, touches the x-axis at ¡2 and cuts the y-axis at (0, 8) c the x-axis at 2, the y-axis at ¡4 and passes through (1, ¡1) and (¡1, ¡21).

INVESTIGATION 3

QUARTIC GRAPHS

There are considerably more possible factor types to consider for quartic functions. We will consider quartics containing certain types of factors. GRAPHING

PACKAGE What to do: 1 Experiment with quartics which have: a four different real linear factors b a squared real linear factor and two different real linear factors c two squared real linear factors d a cubed factor and one real linear factor e a real linear factor raised to the fourth power f one real quadratic factor with ¢ < 0 and two real linear factors g two real quadratic factors each with ¢ < 0.

From Investigation 3 you should have discovered that: For a quartic polynomial in which a is the coefficient of x4 :

²

I If a > 0 the graph opens upwards. I If a < 0 the graph opens downwards. ²

If a quartic with a > 0 is fully factorised into real linear factors, for: I

a single factor (x ¡ ®), the graph cuts the x-axis at ® e.g.

I

a squared factor (x ¡ ®)2 , the graph touches the x-axis at ® e.g.

y

a

y

a x

x

I

a cubed factor (x ¡ ®)3 , the graph cuts the x-axis at ®, but is ‘flat’ at ® e.g.

y

I

a quadruple factor (x ¡ ®)4 , the graph touches the x-axis but is ‘flat’ at that point

a

e.g.

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²

²

If a quartic with a > 0 has one real quadratic factor with ¢ < 0 we could have:

If a quartic with a > 0 has two real quadratic factors both with ¢ < 0 we have:

y

The graph does not meet the x-axis at all.

y

x

x

Example 33 The graph touches the x-axis at ¡1 and cuts it at ¡3 and 3, so y = a(x + 1)2 (x + 3)(x ¡ 3) But when x = 0, y = ¡3

Find the equation of the quartic with graph: y -1

-3

3

) ¡3 = a(1)2 (3)(¡3) ) ¡3 = ¡9a ) a = 13

x

-3

) y = 13 (x + 1)2 (x + 3)(x ¡ 3)

Example 34 Find the quartic which touches the x-axis at 2, cuts it at ¡3, and also passes through (1, ¡12) and (3, 6). (x ¡ 2)2 is a factor as the graph touches the x-axis at 2. (x + 3) is a factor as the graph cuts the x-axis at ¡3. So P (x) = (x ¡ 2)2 (x + 3)(ax + b) where a and b are constants. ) (¡1)2 (4)(a + b) = ¡12 ) 12 (6)(3a + b) = 6

Now P (1) = ¡12, and P (3) = 6,

i.e., a + b = ¡3 ...... (1) i.e., 3a + b = 1 ...... (2)

Solving (1) and (2) simultaneously gives a = 2, b = ¡5 ) P (x) = (x ¡ 2)2 (x + 3)(2x ¡ 5)

EXERCISE 7E.2 1 Find the equation of the quartic with graph: a b y -3

2

c

y -1

x We_

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2 Match the given graphs to the corresponding quartic functions: b y = ¡2(x ¡ 1)2 (x + 1)(x + 3) a y = (x ¡ 1)2 (x + 1)(x + 3) d y = (x ¡ 1)(x + 1)2 (x ¡ 3) c y = (x ¡ 1)(x + 1)2 (x + 3) f y = ¡(x ¡ 1)(x + 1)(x ¡ 3)2 e y = ¡ 13 (x ¡ 1)(x + 1)(x + 3)2 A

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3 Find the equation of the quartic whose graph: a cuts the x-axis at ¡4 and 12 , touches it at 2, and passes through the point (1, 5) b touches the x-axis at 23 and ¡3, and passes through the point (¡4, 49) c cuts the x-axis at § 12 and §2, and passes through the point (1, ¡18) d touches the x-axis at 1, cuts the y-axis at ¡1, and passes through the points (¡1, ¡4) and (2, 15).

DISCUSSION

GENERAL POLYNOMIALS

6 0 What happens to P (x) = an xn + an¡1 xn¡1 + ::::: + a1 x + a0 , an = a polynomial of degree n, n 2 N as jxj ! 1, i.e., as x ! ¡1 and as x ! +1? Notice that as jxj ! 1, the term an xn dominates the value of P (x) and the values of the other terms become insignificant. P (x) ! +1, as x ! +1 and P (x) ! +1, as x ! ¡1.

So, if an > 0 and n is even,

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² ² ²

Discuss the cases where:

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We have already seen that every real cubic polynomial must cut the x-axis at least once, and so has at least one real zero. If the exact value of the zero is difficult to find, we can use technology to help us. We can then factorise the cubic as a linear factor times a quadratic, and if necessary use the quadratic formula to find the other zeros. This method is particularly useful if we have one rational zero and two irrational zeros that are radical conjugates.

Example 35 Find all zeros of P (x) = 3x3 ¡ 14x2 + 5x + 2. Using the calculator we search for any rational zero. In this case x ¼ 0:666 667 or 0:6 indicates x = 23 is a zero and ) (3x ¡ 2) is a factor So, 3x3 ¡ 14x2 + 5x + 2 = (3x ¡ 2)(x2 + ax ¡ 1) = 3x3 + [3a ¡ 2]x2 + [¡3 ¡ 2a]x + 2 3a ¡ 2 = ¡14 and ¡3 ¡ 2a = 5 ) 3a = ¡12 and ¡2a = 8 ) a = ¡4 ) P (x) = (3x ¡ 2)(x2 ¡ 4x ¡ 1) p which has zeros 23 and 2 § 5 fquadratic formulag

Equating coefficients:

TI C GRAPHING PACKAGE

Example 36 Find all roots of 6x3 + 13x2 + 20x + 3 = 0 x ¼ ¡0:166 666 67 = 16 is a zero, so (6x + 1) is a factor. ) (6x + 1)(x2 + ax + 3) = 0 for some constant a. Equating coefficients of x2 : 1 + 6a = 13 ) 6a = 12 ) a = 2 Equating coefficients of x: a + 18 = 20 X p 2 ) (6x + 1)(x + 2x + 3) = 0 and x = ¡ 16 or ¡1 § i 2 fquadratic formulag For a quartic polynomial P (x) we first need to establish if there are any x-intercepts at all. If there are not then the polynomial must have four complex zeros. If there are x-intercepts then we can try to identify linear or quadratic factors.

EXERCISE 7E.3

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all zeros of: x3 ¡ 3x2 ¡ 3x + 1 2x3 ¡ 3x2 ¡ 4x ¡ 35 4x4 ¡ 4x3 ¡ 25x2 + x + 6

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1 Find a c e

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2 Find a c e

the roots of: x3 + 2x2 + 3x + 6 = 0 x3 ¡ 6x2 + 12x ¡ 8 = 0 x4 ¡ x3 ¡ 9x2 + 11x + 6 = 0

3 Factorise into linear factors: a x3 ¡ 3x2 + 4x ¡ 2 c 2x3 ¡ 9x2 + 6x ¡ 1 e 4x3 ¡ 8x2 + x + 3 g 2x4 ¡ 3x3 + 5x2 + 6x ¡ 4

b d f

2x3 + 3x2 ¡ 3x ¡ 2 = 0 2x3 + 18 = 5x2 + 9x 2x4 ¡ 13x3 + 27x2 = 13x + 15

b d f h

x3 + 3x2 + 4x + 12 x3 ¡ 4x2 + 9x ¡ 10 3x4 + 4x3 + 5x2 + 12x ¡ 12 2x3 + 5x2 + 8x + 20

4 The following cubics will not factorise. Find their zeros using technology. b x3 + x2 ¡ 7x ¡ 8 a x3 + 2x2 ¡ 6x ¡ 6

F

THEOREMS FOR REAL POLYNOMIALS

The following theorems are formal statements of discoveries we have made:

²

Unique Factorisation theorem Every real polynomial of degree n can be factorised into n complex linear factors, some of which may be repeated. Factor theorem k is a zero of P (x) , (x ¡ k) is a factor of P (x). Every real polynomial can be expressed as a product of real linear and real irreducible quadratic factors (where ¢ < 0). If p + qi (q 6= 0) is a zero of a real polynomial then its complex conjugate p ¡ qi is also a zero. Every real polynomial of odd degree has at least one real zero. All real polynomials of degree n have n zeros, some of which may be repeated. These zeros are real and/or complex zeros that occur in conjugate pairs.

² ² ² ² ²

Example 37 If ¡3 + i is a zero of P (x) = ax3 + 9x2 + ax ¡ 30 where a is real, find a and hence find all zeros of the cubic. As P (x) is real, both ¡3 + i and ¡3 ¡ i are zeros. These have sum of ¡6 and product of (¡3 + i)(¡3 ¡ i) = 10, so the zeros ¡3 § i come from the quadratic x2 + 6x + 10. Consequently, ax3 + 9x2 + ax ¡ 30 = (x2 + 6x + 10)(ax ¡ 3). -30

axC

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To find a we equate coefficients of x2 and x ) 9 = 6a ¡ 3 and a = 10a ¡ 18 and a = 2 in both cases ) a = 2 and the other two zeros are ¡3 ¡ i and 32 . the linear factor is (ax ¡ 3) i.e., (2x ¡ 3)

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Example 38 One zero of ax3 + [a + 1]x2 + 10x + 15, a 2 R , is purely imaginary. Find a and the zeros of the polynomial. Let the purely imaginary zero be bi, b 6= 0. P (x) is real since its coefficients are all real, and so ¡bi is also a zero. For bi and ¡bi, their sum = 0 and their product = ¡b2 i2 = b2 ) these two zeros come from x2 + b2 . 15 So, ax3 + [a + 1]x2 + 10x + 15 = (x2 + b2 )(ax + 2 ) b axC

·

= ax3 +

15

¸

15 2 x + b2 ax + 15 b2

15 ...... (1) and b2 a = 10 ...... (2) b2 ) b2 a + b2 = 15 fusing (1)g ) 10 + b2 = 15 fusing (2)g p ) b2 = 5 and so b = § 5

Consequently a + 1 =

In (2), as b2 = 5, 5a = 10 )

a = 2.

15 or 2x + 3 b2 p a = 2 and the zeros are §i 5, ¡ 32 .

The linear factor is ax + )

EXERCISE 7F 1 Find all third degree real polynomials with zeros of ¡ 12 and 1 ¡ 3i. 2 p(x) is a real cubic polynomial in which p(1) = p(2 + i) = 0 and p(0) = ¡20. Find p(x) in expanded form. 3 2 ¡ 3i is a zero of P (z) = z 3 + pz + q where p and q are real. Using conjugate pairs, find p and q and the other two zeros. Check your answer by solving for p and q using P (2 ¡ 3i) = 0. 4 3 + i is a root of z 4 ¡ 2z 3 + az 2 + bz + 10 = 0, where a and b are real. Find a and b and the other roots of the equation. 5 One zero of P (z) = z 3 + az 2 + 3z + 9 is purely imaginary. If a is real, find a and hence factorise P (z) into linear factors. 6 At least one zero of P (x) = 3x3 + kx2 + 15x + 10 is purely imaginary. Given that k is real, find k and hence resolve P (x) into a product of linear factors.

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7 A scientist working for Crash Test Barriers, Inc. is trying to design a crash test barrier whose ideal characteristics are shown graphically below. The independent variable t is the time after impact, measured in milliseconds, such that 0 6 t 6 700. The

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dependent variable is the distance that the barrier has been depressed because of the impact, measured in millimetres.

¦¡(t) 120 100 80 60 40 20

t (ms) 200

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a The equation for this graph is of the form f (t) = kt(t ¡ a)2 , 0 6 t 6 700. From the graph, what is the value of a? What does it represent? b If the ideal crash barrier is depressed by 85 mm after 100 milliseconds, find the value of k, and hence find the equation of the graph given. c What is the maximum amount of depression, and when does it occur? 8 In the last year (starting 1st January), the volume of water (in megalitres) in a particular dam after t months could be described by the model V (t) = ¡t3 + 30t2 ¡ 131t + 250. The dam authority rules that if the volume falls below 100 ML, irrigation is prohibited. During which months, if any, was irrigation prohibited in the last twelve months? Include in your answer a neat sketch of xm any graphs you may have used. 10 m 9 A ladder of length 10 metres is leaning up against a wall so that it is just touching a cube shaped box of edge length one metre, that is resting on the ground against the wall. What height up the wall does the ladder reach?

1m

REVIEW SET 7A 1 Find real numbers a and b such that: a a + ib = 4 b (1 ¡ 2i)(a + bi) = ¡5 ¡ 10i

c (a + 2i)(1 + bi) = 17 ¡ 19i

2 If z = 3 + i and w = ¡2 ¡ i, find in simplest form: z¤ a 2z ¡ 3w b w

c

z3

3 Find the exact values of the real and imaginary parts of z if z =

p 3 p + 3 i+ 3

4 Find a complex number z such that 2z ¡ 1 = iz ¡ i. Write your answer in the form z = a + bi where a, b 2 R . 5 Prove that zw¤ ¡ z ¤ w is purely imaginary or zero for all complex numbers z and w. z+1 6 Given w = ¤ where z = a+bi, a, b 2 R , write w in the form x+yi where z +1 x, y 2 R and hence determine the conditions under which w is purely imaginary. (3x3 + 2x ¡ 5)(4x ¡ 3)

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8 Carry out the following divisions:

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7 Expand and simplify:

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9 State and prove the Remainder theorem. 10 ¡2 + bi is a solution to z 2 + az + [3 + a] = 0. Find a and b given that they are real. 11 Find all zeros of 2z 4 ¡ 5z 3 + 13z 2 ¡ 4z ¡ 6. 12 Factorise z 4 + 2z 3 ¡ 2z 2 + 8 into linear factors.

p p 13 Find a quartic polynomial with rational coefficients having 2 ¡ i 3 and 2 + 1 as two of its zeros. 14 If f (x) = x3 ¡ 3x2 ¡ 9x + b has (x ¡ k)2 as a factor, show that there are two possible values of k. For each of these two values of k, find the corresponding value for b and hence solve f (x) = 0. 15 Find k if the line with equation y = 2x + k x2 + y 2 + 8x ¡ 4y + 2 = 0.

does not meet the circle with equation

16 When P (x) = xn + 3x2 + kx + 6 is divided by x + 1 the remainder is 12. When P (x) is divided by x ¡ 1 the remainder is 8. Find k and n given that 34 < n < 38. 17 If ® and ¯ are two of the roots of x3 ¡ x + 1 = 0, show that ®¯ is a root of x3 + x2 ¡ 1 = 0. Hint: Let x3 ¡ x + 1 = (x ¡ ®)(x ¡ ¯)(x ¡ °).

REVIEW SET 7B p 5 1 Find x, y 2 Z such that z = x + yi satisfies the equation 3 z = ¡ 3 ¡ 2i. 2¡i p 2 Without using a calculator, find 5 ¡ 12i. Check your answer using a calculator. 3 Prove that zw¤ ¡ z ¤ w is purely imaginary or zero for all complex numbers z or w. 4 Prove that if z is a complex number then both z + z ¤ and zz ¤ are real. 5 If z = 4 + i and w = 3 ¡ 2i find 2w¤ ¡ iz: 2 ¡ 3i 6 Find rationals a and b such that = 3 + 2i: 2a + bi 7 a + ai is a root of x2 ¡ 6x + b = 0 where a, b 2 R . Explain why b has two possible values. Find a in each case. 8 Find the remainder when x47 ¡ 3x26 + 5x3 + 11 is divided by x + 1. 9 A quartic polynomial P¡(x) has graph y = P¡(x) which touches the x-axis at (¡2, 0), cuts it at (1, 0), cuts the y-axis at (0, 12), and passes through (2, 80). Find an expression for P¡(x) in factored form and hence sketch the graph of y = P¡(x). 10 If P (x) has remainder 2 when divided by x ¡ 3 and remainder ¡13 when divided by x + 2, find the remainder when P (x) is divided by x2 ¡ x ¡ 6. p 11 Find all polynomials of least degree with rational coefficients that have 3 ¡ i 2 and p 1 ¡ 2 as two zeros.

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12 Factorise 2z 3 + z 2 + 10z + 5 as a product of linear factors.

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13 Find the general form of all polynomials of least degree which are real and have zeros of 2 + i and ¡1 + 3i. 14 3 ¡ 2i is a zero of z 4 + kz 3 + 32z + 3k ¡ 1, where k is real. Find k and all zeros of the quartic. 15 Find all the zeros of the polynomial z 4 + 2z 3 + 6z 2 + 8z + 8 given that one of the zeros is purely imaginary. 16 When a polynomial P (x) is divided by x2 ¡ 3x + 2 the remainder is 2x + 3. Find the remainder when P (x) is divided by x ¡ 2.

REVIEW SET 7C 1 Find real numbers x and y such that (3x + 2yi)(1 ¡ i) = (3y + 1)i ¡ x. 2 Solve the equation: z 2 + iz + 10 = 6z 3 Prove that zw¤ + z ¤ w is real for all complex numbers z and w. 4 Find real x and y such that: a x + iy = 0 b (3 ¡ 2i)(x + i) = 17 + yi

c

(x + iy)2 = x ¡ iy

5 z and w are non-real complex numbers with the property that both z + w and zw are real. Prove that z ¤ = w. p 2 z= 6 Find z if + 2 + 5i 3 ¡ 2i 7 Find the remainder when 2x17 + 5x10 ¡ 7x3 + 6 is divided by x ¡ 2. 8 5 ¡ i is a zero of 2z 3 + az 2 + 62z + [a ¡ 5], where a is real. Find a and the other two zeros. 9 Find, in general form, all polynomials of least degree which are real, and have zeros of: p b 1 ¡ i and ¡3 ¡ i: a i 2 and 12 10 P (x) = 2x3 +7x2 +kx¡k is the product of 3 linear factors, 2 of which are identical. Show that k can take 3 distinct values and resolve P (x) into linear factors when k takes the largest of these 3 values. 11 Find all roots of 2z 4 ¡ 3z 3 + 2z 2 = 6z + 4. 12 Suppose k is real. For what values of k does z 3 + az 2 + kz + ka = 0 have: a one real root b 3 real roots? 13 3x + 2 and x ¡ 2 are factors of 6x3 + ax2 ¡ 4ax + b. Find a and b. 14 Find the exact values of k for which the line y = x ¡ k is a tangent to the circle with equation (x ¡ 2)2 + (y + 3)2 = 4.

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15 Find the quotient and remainder when x4 +3x3 ¡7x2 +11x¡1 is divided by x2 +2. Hence, find a and b for which x4 + 3x3 ¡ 7x2 + (2 + a)x + b is exactly divisible by x2 + 2.

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8

Counting and the binomial expansion Contents:

The product principle Counting paths Factorial notation Permutations Combinations Binomial expansions The general binomial expansion

A B C D E F G

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Review set 8A Review set 8B

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COUNTING AND THE BINOMIAL EXPANSION (Chapter 8)

OPENING PROBLEM At an IB Mathematics Teachers’ Conference there are 273 delegates present. The organising committee consists of 10 people. ²

If each committee member shakes hands with every other committee member, how many handshakes take place? Can a 10-sided convex polygon be used to solve this problem?

²

If all 273 delegates shake hands with all other delegates, how many handshakes take place now?

The Opening Problem is an example of a counting problem. The following exercises will help us to solve counting problems without having to list and count the possibilities one by one. To do this we will examine: ² the product principle ² counting permutations ² counting combinations

A

THE PRODUCT PRINCIPLE

Suppose that there are three towns A, B and C and that 4 different roads could be taken from A to B and two different roads from B to C. A

We can show this in a diagram:

C B

How many different pathways are there from A to C going through B?

If we take road 1, there are two alternative roads to complete our trip. If we take road 2, there are two alternative roads road 1 road 2 to complete our trip, ...... and so on. So, there are 2 + 2 + 2 + 2 = 4 £ 2 different pathways.

A

C B

Notice that the 4 corresponds to the number of roads from A to B and the 2 corresponds to the number of roads from B to C. road 1 road 2

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A

Similarly, for:

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there would be 4 £ 2 £ 3 = 24 different pathways from A to D passing through B and C.

THE PRODUCT PRINCIPLE

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If there are m different ways of performing an operation and for each of these there are n different ways of performing a second independent operation, then there are mn different ways of performing the two operations in succession.

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The product principle can be extended to three or more successive operations.

Example 1 P

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It is possible to take five different paths from Pauline’s to Quinton’s, 4 different paths from Quinton’s to Reiko’s and 3 different paths from Reiko’s to Sam’s. How many different pathways could be taken from Pauline’s to Sam’s via Quinton’s and Reiko’s?

The total number of different pathways = 5 £ 4 £ 3 = 60. fproduct principleg

EXERCISE 8A 1 The illustration shows the possible map routes for a bus service which goes from P to S through both Q and R. P How many different routes are possible? 2

Q

R

S

It is decided to label the vertices of a rectangle with the letters A, B, C and D. In how many ways is this possible if: a they are to be in clockwise alphabetical order b they are to be in alphabetical order c they are to be in random order?

3 The figure alongside is box-shaped and made of wire. An ant crawls along the wire from A to B. How many different paths of shortest length lead from A to B?

B

A

4 In how many different ways can the top two positions be filled in a table tennis competition of 7 teams? 5 A football competition is organised between 8 teams. In how many ways is it possible to fill the top 4 places in order of premiership points obtained? 6 How many 3-digit numbers can be formed using the digits 2, 3, 4, 5 and 6: a as often as desired b once only? 7 How many different alpha-numeric plates for motor car registration can be made if the first 3 places are English alphabet letters and those remaining are 3 digits from 0 to 9?

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8 In how many ways can: a 2 letters be mailed into 2 mail boxes c 4 letters be mailed into 3 mail boxes?

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B

COUNTING PATHS

Consider the following road system leading from P to Q:

B

From A to Q there are 2 paths. P From B to Q there are 3 £ 2 = 6 paths. From C to Q there are 3 paths. Thus, from P to Q there are 2 + 6 + 3 = 11 paths. I

Notice that:

G

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C

when going from B to G, we go from B to E and then from E to G, and we multiply the possibilities, when going from P to Q, we must first go from P to A, or P to B or P to C, and we add the possibilities.

I Consequently:

D

A

² ²

the word and suggests multiplying the possibilities the word or suggests adding the possibilities.

Example 2 C

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How many different paths lead from P to Q?

E D

Going

from P to A to or from P to D to or from P to D to So, we have 6 + 2 + 4 = 12

Q

F

P

I

H

G

B to C to Q there are 2 £ 3 = 6 paths E to F to Q there are 2 paths G to H to I to Q there are 2 £ 2 = 4 paths. different paths.

EXERCISE 8B 1 How many different paths lead from P to Q? a b Q

P

P

c

d Q

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C

FACTORIAL NOTATION

In problems involving counting, products of consecutive positive integers are common. For example, 8£7£6 or 6 £ 5 £ 4 £ 3 £ 2 £ 1.

FACTORIAL NOTATION For convenience, we introduce factorial numbers to represent the products of consecutive positive integers. For example, the product 6 £ 5 £ 4 £ 3 £ 2 £ 1 can be written as 6! .

for n > 1, n! is the product of the first n positive integers.

In general,

n! = n(n ¡ 1)(n ¡ 2)(n ¡ 3):::: £ 3 £ 2 £ 1 n! is read “n factorial”. Notice that 8 £ 7 £ 6 can be written using factorial numbers only as 8£7£6£5£4£3£2£1 8! 8£7£6= = 5£4£3£2£1 5!

PROPERTIES OF FACTORIAL NUMBERS n! = n £ (n ¡ 1)! for n > 1

The factorial rule is

which can be extended to n! = n(n ¡ 1)(n ¡ 2)! and so on. Using the factorial rule with n = 1, we have 1! = 1 £ 0! 0! = 1

We hence define

Example 3 a 4!

What integer is equal to:

7! ? 4! £ 3!

b

5! 3!

b

5! 5£4£3£2£1 = = 5 £ 4 = 20 3! 3£2£1

a

4! = 4 £ 3 £ 2 £ 1 = 24

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7! 7£6£5£4£3£2£1 = = 35 4! £ 3! 4£3£2£1£3£2£1

c

Example 4

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10 £ 9 £ 8 £ 7 4£3£2£1

10 £ 9 £ 8 £ 7 £ 6 £ 5 £ 4 £ 3 £ 2 £ 1 10! = 6£5£4£3£2£1 6!

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10 £ 9 £ 8 £ 7 10 £ 9 £ 8 £ 7 £ 6 £ 5 £ 4 £ 3 £ 2 £ 1 10! = = 4£3£2£1 4£3£2£1£6£5£4£3£2£1 4! £ 6!

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Express in factorial form: a 10 £ 9 £ 8 £ 7

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Example 5 Write the following sums and differences as a product by factorising: a 8! + 6! b 10! ¡ 9! + 8! a

10! ¡ 9! + 8! = 10 £ 9 £ 8! ¡ 9 £ 8! + 8! = 8!(90 ¡ 9 + 1) = 8! £ 82

b

8! + 6! = 8 £ 7 £ 6! + 6! = 6!(8 £ 7 + 1) = 6! £ 57

Example 6 7! ¡ 6! 6

Simplify

7! ¡ 6! 6 7 £ 6! ¡ 6! = 6 1 6!(7 ¡ 1) = 6 1

using factorisation.

= 6!

EXERCISE 8C 1 Find n! for n = 0, 1, 2, 3, ::::, 10. 2 Simplify without using a calculator: 6! 6! 6! a b c 5! 4! 7! 3 Simplify:

n! (n ¡ 1)!

a

13 £ 12 £ 11 3£2£1

d

100! 99!

e

(n + 2)! n!

b

4 Express in factorial form: a 7£6£5

4! 6!

d

f

7! 5! £ 2!

(n + 1)! (n ¡ 1)!

c

b

10 £ 9

c

11 £ 10 £ 9 £ 8 £ 7

e

1 6£5£4

f

4£3£2£1 20 £ 19 £ 18 £ 17

5 Write as a product using factorisation: b 11! ¡ 10! a 5! + 4! e 9! + 8! + 7! f 7! ¡ 6! + 8!

c g

d h

6! + 8! 12! ¡ 2 £ 11!

12! ¡ 10! 3 £ 9! + 5 £ 8!

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D

PERMUTATIONS

A permutation of a group of symbols is any arrangement of those symbols in a definite order. For example, BAC is a permutation on the symbols A, B and C in which all three of them are used. We say the symbols are “taken 3 at a time”. Notice that ABC, ACB, BAC, BCA, CAB, CBA are all the different permutations on the symbols A, B and C taken 3 at a time. In this exercise we are concerned with the listing of all permutations, and then learning to count how many permutations there are without having to list them all.

Example 7 List all the permutations on the symbols P, Q and R when they are taken: a 1 at a time b 2 at a time c 3 at a time. a

b

P, Q, R

PQ PR

QP QR

c

RP RQ

PQR PRQ

QPR QRP

RPQ RQP

Example 8 List all permutations on the symbols W, X, Y and Z taken 4 at a time. WXYZ XWYZ YWXZ ZWXY

WXZY XWZY YWZX ZWYX

WYXZ XYWZ YXWZ ZXWY

WYZX XYZW YXZW ZXYW

WZXY XZYW YZWX ZYWX

WZYX XZWY YZXW ZYXW

i.e., 24 of them.

For large numbers of symbols listing the complete set of permutations is absurd. However, we can still count them in the following way. In Example 8 there were 4 positions to fill: 1st 2nd 3rd 4th

In the 1st position, any of the 4 symbols could be used. 4

This leaves any of 3 symbols to go in the 2nd position, which leaves any of 2 symbols to go in the 3rd position.

1st 2nd 3rd 4th 4

3

2

1st 2nd 3rd 4th

The remaining symbol must go in the 4th position.

4

3

2

1

1st 2nd 3rd 4th

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Example 9 If a chess association has 16 teams, in how many different ways could the top 8 positions be filled on the competition ladder? Any of the 16 teams could fill the ‘top’ position. Any of the remaining 15 teams could fill the 2nd position. Any of the remaining 14 teams could fill the 3rd position. .. . Any of the remaining 9 teams could fill the 8th position. i.e.,

16 1st

15 2nd

14 3rd

13 4th

12 5th

11 6th

10 7th

9 8th

) total number = 16 £ 15 £ 14 £ 13 £ 12 £ 11 £ 10 £ 9 = 518 918 400

Example 10 Suppose you have the alphabet blocks A, B, C, D and E and they are placed in a row. For example you could have: D A E C B a How many different permutations could you have? b How many permutations end in C? c How many permutations have the form ... A ... B ... ? d How many begin and end with a vowel, i.e., A or E? a There are 5 letters taken 5 at a time. ) total number = 5 £ 4 £ 3 £ 2 £ 1 = 120. b

4 3 2 1 1

C must be in the last position (1 way) and the other 4 letters could go into the remaining 4 places in 4! ways.

any others here C here

c

) total number = 1 £ 4! = 24 ways.

3 1 2 1 1 A

A goes into 1 place, B goes into 1 place and the remaining 3 letters go into the remaining 3 places in 3! ways.

B

) total number = 1 £ 1 £ 3! = 6 ways. d

2 3 2 1 1

A or E could go into the 1st position, and after that one is placed, the other one goes into the last position.

A or E remainder of A or E

The remaining 3 letters could be arranged in 3! ways in the 3 remaining positions.

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Example 11 There are 6 different books arranged in a row on a shelf. In how many ways can two of the books, A and B be together? Method A B B A £ A £ B £ £ £ £ £ £ £ £ £ £ £ £

1: We could £ £ £ £ £ £ B £ £ A £ £ A B £ B A £ £ A B £ B A £ £ A £ £ B

have any of the following locations for A and B 9 £ > > > £ > > If we consider any one > > £ > > of these, the remaining > £ > > = 4 books could be placed £ 10 of these £ in 4! different orderings > > > £ > ) total number of ways > > £ > > = 10 £ 4! = 240: > > B > > ; A

Method 2: A and B can be put together in 2! ways (i.e., AB or BA). Now consider this pairing as one book (effectively tying a string around them) which together with the other 4 books can be ordered in 5! different ways. ) total number = 2! £ 5! = 240:

EXERCISE 8D 1 List the set of all permutations on the symbols W, X, Y and Z taken: a 1 at a time b two at a time c three at a time. Note: Example 8 has them taken 4 at a time. 2 List the set of all permutations on the symbols A, B, C, D and E taken: a 2 at a time b 3 at a time. 3 In how many ways can: a 5 different books be arranged on a shelf b 3 different paintings, from a collection of 8, be chosen and hung in a row c a signal consisting of 4 coloured flags be made if there are 10 different flags to choose from? 4 Suppose you have 4 different coloured flags. How many different signals could you make using: a 2 flags only b 3 flags only c 2 or 3 flags? 5 How many different permutations of the letters A, B, C, D, E and F are there if each letter can be used once only? How many of these: a end in ED b begin with F and end with A c begin and end with a vowel (i.e., A or E)?

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6 How many 3-digit numbers can be constructed from the digits 1, 2, 3, 4, 5, 6 and 7 if each digit may be used: a as often as desired b only once c once only and the number is odd?

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7 In how many ways can 3 boys and 3 girls be arranged in a row of 6 seats? In how many of these ways do the boys and girls alternate? 8 3-digit numbers are constructed from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 using each digit at most once. How many such numbers: a can be constructed b end in 5 c end in 0 d are divisible by 5? 9 In how many ways can 5 different books be arranged on a shelf if: a there are no restrictions b books X and Y must be together c books X and Y are never together? 10 A group of 10 students sit randomly in a row of 10 chairs. In how many ways can this be done if: a there are no restrictions b 3 students A, B and C are always seated together? 11 How many three-digit numbers, in which no two digits are the same, can be made using the digits 0, 1, 3, 5, 8 if: a there are no restrictions b the numbers must be less than 500 c the numbers must be even and greater than 300? 12 How many different arrangements of four letters chosen from the letters of the word MONDAY are possible if: a there are no restrictions b at least one vowel (A or O) must be used c no two vowels are adjacent? 13 Nine boxes are each labelled with a different whole number from 1 to 9. Five people are allowed to take one box each. In how many different ways can this be done if: a there are no restrictions b the first three people decide that they will take even numbered boxes? 14 Alice has booked ten adjacent front-row seats for a basketball game for herself and nine friends. Altogether, there are five boys and five girls. a Assuming they all arrive, how many different arrangements are there if: i there are no restrictions ii boys and girls are to sit alternately? b Due to a severe snowstorm, only five of Alice’s friends are able to join her for the game. How many different ways are there of seating in the 10 seats if: i there are no restrictions ii any two of Alice’s friends are to sit next to her?

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15 At a restaurant, a rectangular table seats eight people, four on each of the longer sides. Eight diners sit at the table. How many different seating arrangements are there if: a there are no restrictions b two particular people wish to sit directly opposite each other c two particular people wish to sit on the same side of the table, next to each other?

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INVESTIGATION 1

PERMUTATIONS IN A CIRCLE

There are 6 permutations on the symbols A, B and C in a line. These are: ABC ACB BAC BCA CAB CBA. However in a circle there are only 2 different permutations on these 3 symbols. These are: B

C

and

A C

Notice that

as they are the only possibilities with different right-hand and lefthand neighbours.

A B A

B

C

C

A

are the same cyclic permutations.

B B

C

A

What to do:

1 Draw diagrams showing different cyclic permutations for: a one symbol: A b two symbols: A and B c three symbols: A, B and C d four symbols: A, B, C and D 2 Copy and complete:

Number of symbols Permutations in a line Permutations in a circle 1 2 3 6 = 3! 2 = 2! 4

3 If there are n symbols to be permuted in a circle, how many different orderings are possible?

E

COMBINATIONS A combination is a selection of objects without regard to order or arrangement.

For example, the possible teams of 3 people selected from A, B, C, D and E are ABC BCD CDE

ABD BCE

ABE BDE

ACD

ACE

ADE

i.e., 10 different combinations.

Crn is the number of combinations on n distinct or different symbols taken r at a time. the number up for selection the number of positions needed to be filled. ¡ ¢ may also be written as n Cr or as the binomial coefficient nr . Crn

Crn

From the teams example above we know that C35 = 10:

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However, we know the number of permutations of three people from the 5 possibilities is 5 £ 4 £ 3 = 60, so why is this answer 6 or 3! times larger than C35 ?

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This can be seen if we consider one of these teams, ABC say. There are 3! ways in which the members of team ABC can be placed in a definite order, i.e., ABC, ACB, BAC, BCA, CAB, CBA and if this is done for all 10 possible teams we get all possible permutations of the 5 people taken 3 at a time. So, 5 £ 4 £ 3 = C35 £ 3!

) C35 =

5£4£3 3£2£1

or

5! . 3! £ 2!

n (n ¡ 1)(n ¡ 2) .... (n ¡ r + 3) (n ¡ r + 2) (n ¡ r + 1) n! = In general, Crn = r (r ¡ 1) (r ¡ 2) .... 3 2 1 r!(n ¡ r)! | {z } | {z } Factor form Factorial form Values of Crn can be calculated from your calculator. For example, to find C310 : TI-83:

Press 10 MATH

3 to select 3:n Cr from the PRB menu, then 3 ENTER

Casio: Press 10 OPTN

F3 (PROB) F3 (n Cr) 3

F6

EXE

Example 12 How many different teams of 4 can be selected from a squad of 7 if: a there are no restrictions b the teams must include the captain? a

There are 7 players up for selection and we want any 4 of them. This can be done in C47 = 35 ways.

b

If the captain must be included and we need any 3 of the other 6, this can be done in C11 £ C36 = 20 ways.

Example 13 A committee of 4 is chosen from 7 men and 6 women. How many different committees can be chosen if: a there are no restrictions b there must be 2 of each sex c at least one of each sex is needed? a

For no restrictions there are 7 + 6 = 13 people up for selection and we want any 4 of them. ) total number = C413 = 715:

b

The 2 men can be chosen in C27 ways and the 2 women can be chosen in C26 ways. ) total number = C27 £ C26 = 315: Total number = number with (3 M and 1 W) or (2 M and 2 W) or (1 M and 3 W) = C37 £ C16 + C27 £ C26 + C17 £ C36 = 665

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total number = C413 ¡ C47 £ C06 ¡ C07 £ C46 :

Alternatively,

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c

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EXERCISE 8E 1 Evaluate using factor form: a C18 b Check each answer using your calculator.

C28

c

C38

d

C68

e

C88 :

2 In question 1 you probably noticed that C28 = C68 . n . Prove using factorial form that this statement is true. In general, Crn = Cn¡r ³ ´ ¡9¢ 7 = 4 3 Find k if k k¡1 : 4 List the different teams of 4 that can be chosen from a squad of 6 (named A, B, C, D, E and F). Check that the formula for Crn gives the total number of teams. 5 How many different teams of 11 can be chosen from a squad of 17? 6 Candidates for an examination are required to do 5 questions out of 9. In how many ways can this be done? If question 1 was compulsory, how many selections would be possible? 7 How many different committees of 3 can be selected from 13? How many of these committees consist of the president and 2 others? 8 How many different teams of 5 can be selected from a squad of 12? How many of these teams contain: a the captain and vice-captain b exactly one of the captain or the vice-captain? 9 A team of 9 is selected from a squad of 15. 3 of the players are certainties who must be included, and another must be excluded because of injury. In how many ways can this be done? 10 In how many ways can 4 people be selected from 10 if: a one person is always in the selection b 2 are excluded from every selection c 1 is always included and 2 are always excluded? 11 A committee of 5 is chosen from 10 men and 6 women. Determine the number of ways of selecting the committee if: a there are no restrictions b it must contain 3 men and 2 women c it must contain all men d it must contain at least 3 men e it must contain at least one of each sex. 12 A committee of 5 is chosen from 6 doctors, 3 dentists and 7 others. Determine the number of ways of selecting the committee if it is to contain: a 2 doctors and 1 dentist b 2 doctors c at least one of the two professions. 13 How many diagonals has a 20-sided convex polygon? 14 There are 12 distinct points A, B, C, D, ..., L on a circle. Lines are drawn between each pair of points. a How many lines i are there in total ii pass through B? b How many triangles i are determined by the lines ii have one vertex B?

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15 How many 4-digit numbers can be constructed where the digits are in ascending order from left to right? Note: You cannot start with 0.

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a Give an example which demonstrates that: C05 £ C46 + C15 £ C36 + C25 £ C26 + C35 £ C16 + C45 £ C06 = C411 : b Copy and complete: n n m + C2m £Cr¡2 + :::: + Cr¡1 £C1n + Crm £C0n = ::::: C0m £Crn + C1m £Cr¡1

17 In how many ways can 12 people be divided into: a two equal groups b three equal groups? A

18 Line A contains 10 points and line B contains 7 points. If all points on line A are joined to all points on line B, determine the maximum number of points of intersection between A and B.

B Q

19 10 points are located on [PQ], 9 on [QR] and 8 on [RP]. All possible lines connecting these 27 points are drawn. Determine the maximum number of points of intersection of these lines which lie within triangle PQR.

F

R

P

BINOMIAL EXPANSIONS The sum a + b is called a binomial as it contains two terms. Any expression of the form (a + b)n is called a power of a binomial.

Consider the following algebraic expansions of the powers (a + b)n . (a + b)1 = a + b (a + b)2 = a2 + 2ab + b2

(a + b)3 = (a + b)(a + b)2 = (a + b)(a2 + 2ab + b2 ) = a3 + 2a2 b + ab2 + a2 b + 2ab2 + b3 = a3 + 3a2 b + 3ab2 + b3

We say that: a2 + 2ab + b2 a3 + 3a2 b + 3ab2 + b3

is the binomial expansion of is the binomial expansion of

(a + b)2 (a + b)3

THE BINOMIAL EXPANSION OF (a¡+¡b)n, n¡>¡4

INVESTIGATION 2 What to do:

1 Expand (a + b)4

in the same way as for (a + b)3 5

2 Similarly, expand algebraically (a + b) expansion of (a + b)4 from 1. 3 Expand (a + b)6

above.

using your answer for the

using your expansion for (a + b)5 .

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4 The (a + b)3 = a3 + 3a2 b + 3ab2 + b3 expansion contains 4 terms: a3 , 3a2 b, 3ab2 and b3 . The coefficients of these terms are: 1 3 3 1

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a What can be said about the powers of a and b in each term of the expansion of (a + b)n for n = 0, 1, 2, 3, 4, 5 and 6? b Write down the triangle of coefficients to row 6:

n=0 n=1 n=2 n=3

1 1 1 1

1 2

3

1 3

.. . etc.

1

row 3

5 This triangle of coefficients is called Pascal’s triangle. Investigate: a the predictability of each row from the previous one b a formula for finding the sum of the numbers in the nth row of Pascal’s triangle. 6 Use your results from 5 to predict the elements of the 7th row of Pascal’s triangle and hence write down the binomial expansion of (a + b)7 . Check your result algebraically by using (a+b)7 = (a+b)(a+b)6 and your results from 3 above. (a + b)4 = a4 + 4a3 b + 6a2 b2 + 4ab3 + b4 = a4 + 4a3 b1 + 6a2 b2 + 4a1 b3 + b4

From Investigation 2 we obtained Notice that:

²

As we look from left to right across the expansion, the powers of a decrease by 1 and the powers of b increase by 1. The sum of the powers of a and b in each term of the expansion is 4. The number of terms in the expansion is 4 + 1 = 5.

² ²

In general, for the expansion of (a + b)n ²

where n = 1, 2, 3, 4, 5, .... :

As we look from left to right across the expansion, the powers of a decrease by 1 whilst the powers of b increase by 1. The sum of the powers of a and b in each term of the expansion is n. The number of terms in the expansion is n + 1.

² ²

The expansion (a + b)3 = a3 + 3a2 b + 3ab2 + b3

can be used to expand other cubes.

Example 14 Using (a + b)3 = a3 + 3a2 b + 3ab2 + b3 , find the binomial expansion of: b (x ¡ 5)3 a (2x + 3)3 a

In the expansion of (a + b)3 3

we substitute a = (2x) and b = (3)

(2x + 3) = (2x) + 3(2x)2 (3) + 3(2x)1 (3)2 + (3)3

)

3

= 8x3 + 36x2 + 54x + 27 on simplifying b

This time we substitute a = (x) and b = (¡5) (x ¡ 5)3 = (x)3 + 3(x)2 (¡5) + 3(x)(¡5)2 + (¡5)3

)

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= x3 ¡ 15x2 + 75x ¡ 125

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Example 15 ¢5 ¡ b binomial expansion of x ¡ x2 .

Find the: a 5th row of Pascal’s triangle a

the 0th row, for (a + b)0 the 1st row, for (a + b)1

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1

b

the 5th row

So, (a + b)5 = a5 + 5a4 b + 10a3 b2 + 10a2 b3 + 5ab4 + b5 ¡ ¢ and we let a = (x) and b = ¡2 x ¡ ¢ ¡ ¢ ¡ ¢2 ¡ ¢3 2 5 5 4 ¡2 ) x ¡ x = (x) + 5(x) x + 10(x)3 ¡2 + 10(x)2 ¡2 x x ¡ ¡2 ¢4 ¡ ¡2 ¢5 + 5(x) x + x = x5 ¡ 10x3 + 40x ¡

80 80 32 + 3¡ 5 x x x

EXERCISE 8F 1 Use the binomial expansion of (a + b)3 a

(x + 1)3

b

to expand and simplify:

(3x ¡ 1)3

(2x + 5)3

c

2 Use (a + b)4 = a4 + 4a3 b + 6a2 b2 + 4ab3 + b4 a

(x ¡ 2)4

b

(2x + 3)4

c

d

to expand and simplify: ¢4 ¢4 ¡ ¡ x + x1 d 2x ¡ x1

a Write down the 6th row of Pascal’s triangle. b Find the binomial expansion of: i (x + 2)6 ii (2x ¡ 1)6

3

4 Expand and simplify: p a (1 + 2)3

b

(1 +

¢3 ¡ 2x + x1

iii

p 4 5)

c

¢6 ¡ x + x1 (2 ¡

p 5 2)

a Expand (2 + x)6 . b Use the expansion of a to find the value of (2:01)6 .

5

6 Expand and simplify (2x + 3)(x + 1)4 . 7 Find the coefficient of:

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b a3 b3 in the expansion of (2a + 3b)6 .

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THE GENERAL BINOMIAL EXPANSION

¡ ¢ ¡ ¢ ¡ ¢ ¡ n ¢ n¡1 ¡n¢ n ab (a + b)n = n0 an + n1 an¡1 b + n2 an¡2 b2 + ::::: + n¡1 + n b ¡n¢ where r = Crn is the binomial coefficient of an¡r br and r = 0, 1, 2, 3, ...., n. The general term, or (r + 1)th term is Tr +1 =

¡n ¢ n ¡ r r a b . r

Example 16 Write down the first 3 and last 2 terms of the expansion of

¢12 ¡ 2x + x1 .

¢12 ¡ ¢ ¡12¢ ¡ ¢ ¡ ¡ ¢ 11 1 10 1 2 2x + x1 = (2x)12 + 12 + :::::: 1 (2x) x + 2 (2x) x ¡ 1 ¢11 ¡12¢ ¡ 1 ¢12 ¡ ¢ + 12 x :::::: + 12 11 (2x) x

Example 17 Find the 7th term of ¡ ¢14 3x ¡ x42 .

a = (3x), b =

¡ ¡4 ¢ x2

and n = 14

¡n¢ n¡r r b , we let r = 6 r a ¡ ¢6 ¡14¢ T7 = 6 (3x)8 ¡4 x2

So, as Tr+1 =

Do not simplify.

)

Example 18 In the expansion of

¡ 2 4 ¢12 x + x , find:

a the coefficient of x6 a = (x2 ), b =

a

¡4¢

b the constant term ¡12¢ 2 12¡r ¡ 4 ¢r r (x ) x ¡12¢ 24¡2r 4r = r x xr ¡12¢ r 24¡3r = r 4 x

)

and n = 12

x

Tr+1 =

b If 24 ¡ 3r = 0 then 3r = 24 ) r=8 ¡ ¢ 8 0 ) T9 = 12 8 4 x

If 24 ¡ 3r = 6 then 3r = 18 ) r=6 ¡ ¢ 6 6 ) T7 = 12 6 4 x

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) the constant term is ¡12¢ 8 or 32 440 320. 8 4

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Example 19 (x + 3)(2x ¡ 1)6 ¡¢ ¡¢ = (x + 3)[(2x)6 + 61 (2x)5 (¡1) + 62 (2x)4 (¡1)2 + ::::] ¡¢ ¡¢ = (x + 3)(26 x6 ¡ 61 25 x5 + 62 24 x4 ¡ ::::)

Find the coefficient of x5 in the expansion of (x + 3)(2x ¡ 1)6 .

(2)

(1)

¡6¢ 4 5 from (1) 2 2 x ¡6¢ 5 5 and ¡3 1 2 x from (2) ¡¢ ¡¢ ) the coefficient of x5 is 62 24 ¡ 3 61 25 = ¡336 So, the terms containing x5 are

EXERCISE 8G 1 Write down the first three and last two terms of the binomial expansion of: ¢15 ¢20 ¡ ¡ a (1 + 2x)11 b 3x + x2 c 2x ¡ x3 2 Without simplifying, find: a the 6th term of (2x + 5)15 ¢17 ¡ c the 10th term of x ¡ x2

the 4th term of

d

3 Find the coefficient of: a x10 in the expansion of (3 + 2x2 )10 ¢12 ¡ c x12 in the expansion of 2x2 ¡ x1 4 Find the constant term in: ¡ a the expansion of x +

¡ 2 5 ¢9 x +x ¡ 2 1 ¢21 the 9th term of 2x ¡ x

b

¢ 2 15 x2

¢6 ¡ x3 in the expansion of 2x2 ¡ x3

b

¡ the expansion of x ¡

b

¢ 3 9 x2

5

a Write down the first 5 rows of Pascal’s triangle. b What is the sum of the numbers in: i row 1 ii row 2 iii row 3 iv row 4 v row 5? c Copy and complete: The sum of the numbers in row n of Pascal’s triangle is ...... ¡ ¢ ¡ ¢ ¡ ¢ ¡ n ¢ n¡1 ¡n¢ n x d Show that (1 + x)n = n0 + n1 x + n2 x2 + :::: + n¡1 + n x ¡ n ¢ ¡n¢ ¡n¢ ¡n¢ ¡n¢ Hence deduce that 0 + 1 + 2 + :::: + n¡1 + n = 2n

6

a Find the coefficient of x5 in the expansion of (x + 2)(x2 + 1)8

b Find the coefficient of x6 in the expansion of (2 ¡ x)(3x + 1)9 ¢5 ¡ 7 Expand 1 + 2x ¡ x2 in ascending powers of x as far as the term containing x4 . 8

a Show that

¡n¢ 1

= n and

¡n¢ 2

=

n(n ¡ 1) 2

are true statements.

b The third term of (1 + x)n is 36x2 . Find the fourth term.

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c If (1 + kx)n = 1 ¡ 12x + 60x2 ¡ ::::: , find the values of k and n.

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9 Find a if the coefficient of x11 in the expansion of (x2 + 10 From the binomial expansion of (1 + x)n , deduce that: ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ a n0 ¡ n1 + n2 ¡ n3 + ...... + (¡1)n nn = 0 ¡ ¢ ¡2n+1¢ ¡2n+1¢ ¡ ¢ b 2n+1 + 1 + 2 + ...... + 2n+1 = 4n 0 n 11 By considering the binomial expansion of (1 + x)n , find

231

1 10 ) is 15. ax

n X

2r

¡n¢ r :

r=0

12 (x2 ¡ 3x + 1)2 = x4 ¡ 6x3 + 11x2 ¡ 6x + 1 and the sum of its coefficients is 1 ¡ 6 + 11 ¡ 6 + 1 which is 1. What is the sum of the coefficients of (x3 + 2x2 + 3x ¡ 7)100 ? 13 By considering (1 + x)n (1 + x)n = (1 + x)2n , show that ¡n¢2 ¡n¢2 ¡n¢2 ¡n¢2 ¡ ¢2 ¡ ¢ + 1 + 2 + 3 + ...... + nn = 2n 0 n . 14

¡ ¢ = n n¡1 r¡1 . ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ b Hence show that n1 + 2 n2 + 3 n3 + 4 n4 + ...... + n nn = n2n¡1 a Prove that r

¡n¢ r

c Suppose numbers Pr are defined by ¡ ¢ Pr = nr pr (1 ¡ p)n¡r for r = 0, 1, 2, 3, ...... , n. n X

i

Prove that

ii

Pr = 1

r=0

n X

rPr = np

r=1

REVIEW SET 8A 1 Alpha-numeric number plates have two letters followed by four digits. How many plates are possible if: a there are no restrictions b the first letter must be a vowel c no letter or digit may be repeated? 2 Ten points are located on a 2-dimensional plane. If no three points are collinear: a how many line segments joining two points can be drawn b how many different triangles can be drawn by connecting all 10 points with line segments in any possible way? 3 Simplify: 4

a

n! (n ¡ 2)!

b

n! + (n + 1)! n!

a How many committees of five can be selected from eight men and seven women? b How many of the committees contain two men and three women? c How many contain at least one man?

5 Eight people enter a room and each person shakes hands with every other person. How many hand shakes are made?

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6 Use the binomial expansion to find:

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(3x + 2)4

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7 A team of five is chosen from six men and four women. a How many different teams are possible with no restrictions? b How many contain at least one of each sex? 8 The letters P, Q, R, S and T are to be arranged in a row. How many of the possible arrangements: a end with T b begin with P and end with T? 9 Find the coefficient of x3 in the expansion of (2x + 5)6 . ¡ ¢6 10 Find the constant term in the expansion of 2x2 ¡ x1 : 11 The first three terms in the expansion of (1 + kx)n , in ascending powers of x, are 2 1 ¡ 4x + 15 2 x . Find k and n. 12 Eight people enter a room and sit at random in a row of eight chairs. In how many ways can the sisters Cathy, Robyn and Jane sit together in the row? a How many three digit numbers can be formed using the digits 0 to 9? b How many of these numbers are divisible by 5?

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REVIEW SET 8B 1 A team of eight is chosen from 11 men and 7 women. How many different teams are possible if there are: a no restrictions b four of each sex on the team c at least two women on the team? 2 A four digit number is constructed using the digits 0, 1, 2, 3, ...., 9 once only. a How many numbers are possible? b How many are divisible by 5? 3 Use Pascal’s triangle to expand (a + b)6 .

¢6 ¡ Hence, find the binomial expansion of: a (x ¡ 3)6 b 1 + x1 p p 4 Expand and simplify ( 3+2)5 giving your answer in the form a+b 2, a, b 2 Z .

5 Use the expansion of (4 + x)3 to find the exact value of (4:02)3 . ¢8 ¡ 6 Find the constant term in the expansion of 3x2 + x1 . ¢12 ¡ 7 Find the coefficient of x¡6 in the expansion of 2x ¡ x32 . 8 Find the coefficient of x5 in the expansion of (2x + 3)(x ¡ 2)6 . ¢9 ¡ 9 Find the possible values of a if the coefficient of x3 in 2x + ax1 2 ¡ 3x ¡

10 Find the term independent of x in the expansion of

is 288.

¢ 2 6 . x2

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11 Find the possible values of q if the constant terms in the expansions of ¡ 3 ¢8 ¢4 ¡ x + xq3 and x3 + xq3 are equal.

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Chapter

Mathematical induction Contents:

A B C

9

The process of induction The principle of mathematical induction Indirect proof (extension)

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MATHEMATICAL INDUCTION (Chapter 9)

A

THE PROCESS OF INDUCTION

The process of formulating a general result from a close examination of the simplest cases is called mathematical induction. For example,

the the the the

2=2£1 4=2£2 6=2£3 8=2£4

first positive even number is second positive even number is third positive even number is fourth positive even number is

and from these results we induce that the nth positive even number is 2 £ n or 2n. The statement that “the nth positive even number is 2n” is a summary of the observations of the simple cases n = 1, 2, 3, 4 and is a statement which we believe is true. We call such a statement a conjecture or proposition. We need to prove a conjecture before we can regard it as a fact. Consider the following argument for finding the sum of the first n odd numbers: 1 = 1 = 12 1 + 3 = 4 = 22 1 + 3 + 5 = 9 = 32 1 + 3 + 5 + 7 = 16 = 42 1| + 3 +{z 5 + 7 + 9} = 25 = 52 5 of these It seems that “the sum of the first n odd numbers is n2 ”. This pattern may continue or it may not. We require proof of the fact for all positive integers n. A formal statement of our conjecture may be: “ 1| + 3 + 5 +{z7 + 9 + :::::} = n2 n of these

for all n 2 Z + ”

The nth odd number is (2n ¡ 1), so we could also write the proposition as: “1 + 3 + 5 + 7 + 9 + ::::: + (2n ¡ 1) = n2

for all n 2 Z + ”

One direct proof of the proposition is to note that the series is arithmetic with u1 = 1, d = 2 and “n” = n. n (2(1) + (n ¡ 1)2) 2 n = £ 2n 2

fusing

Hence Sn =

n (2u1 + (n ¡ 1)d)g 2

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= n2

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Example 1 By examining the cases n = 1, 2, 3 and 4, make a conjecture about the sum 1 1 1 1 1 + 2£3 + 3£4 + 4£5 + ::::: + n(n+1) : of Sn = 1£2 S1 =

1 1£2

=

1 2

S2 =

1 1£2

+

1 2£3

S3 =

1 1£2

S4 =

1 1£2

²

Note:

=

1 2

1 6

=

+

1 2£3

+

1 3£4

=

2 3

+

1 2£3

+

1 3£4

+

1 4£5

+

2 3

+

1 12

=

=

3 4

3 4

+

1 20

=

4 5

If the result in Example 1 is true, then: 1 1 1 1 1 1£2 + 2£3 + 3£4 + 4£5 + ::::: + 1000£1001 =

From these results we conjecture that: n Sn = : n+1

1000 1001

fcase n = 1000g

The great Swiss mathematician Euler proposed that P (n) = n2 + n + 41 was a formula for generating prime numbers. People who read his statement probably checked it for n = 1, 2, 3, 4, 5, ....., 10 and agreed with him. However, it was found to be incorrect as, for example P (41) = 412 + 41 + 41 = 41(41 + 1 + 1) = 41 £ 43, a composite. So, not all propositions are true.

²

EXERCISE 9A 1 By examining the following using substitutions like n = 1, 2, 3, 4, ....., complete the proposition or conjecture. a The nth term of the sequence 3, 7, 11, 15, 19, ..... is for n = 1, 2, 3, 4, ..... 3n > 1 + 2n for for 11n ¡ 1 is divisible by 2 + 4 + 6 + 8 + 10 + ::::: + 2n = for 1! + 2 £ 2! + 3 £ 3! + 4 £ 4! + ::::: + n £ n! = for 1 2 3 4 n f + + + + ::::: + = for 2! 3! 4! 5! (n + 1)! for g 7n + 2 is divisible by ³ ´ ¢ ¡ ¢ ¡ ¢ ¡ 1 h 1 ¡ 12 1 ¡ 13 1 ¡ 14 ::::: 1 ¡ n+1 = for

b c d e

1 1 1 + + + ::::: to n terms = 2 £ 5 5 £ 8 8 £ 11

i

for

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2 n points are placed inside a triangle. Non-intersecting line segments are drawn connecting the 3 vertices of the triangle and the points within it, to partition the given triangle into smaller triangles. Make a proposition concerning the number of triangles obtained in the general case.

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B

THE PRINCIPLE OF MATHEMATICAL INDUCTION

PROPOSITION NOTATION We use Pn to represent a proposition which is defined for every integer n where n > a, a 2 Z . For example, in the case of Example 1, our proposition Pn is 1 1 1 1 n “ + + ::::: + = ” for all n 2 Z + . 1£2 2£3 3£4 n(n + 1) n+1 Notice that

P1 is “

1 1 1 1 2 = ” and P2 is “ + = ” 1£2 2 1£2 2£3 3

and

Pk is “

1 1 1 1 k + + + ::::: + = ”. 1£2 2£3 3£4 k(k + 1) k+1

THE PRINCIPLE OF MATHEMATICAL INDUCTION Suppose Pn is a proposition which is defined for every integer n > a, a 2 Z . ² Pa is true, and ² Pk+1 is true whenever Pk is true, then Pn is true for all n > a.

Now if

This means that for a = 1, say, if the two above conditions hold, then the truth of P1 implies that P2 is true, which implies that P3 is true, which implies that P4 is true, and so on. The principle of mathematical induction constitutes a formal proof that a particular proposition is true. One can liken the principle of mathematical induction to the domino effect. We imagine an infinite set of dominoes all lined up. Provided that: ² the first one topples to the right, and ² the (k + 1)th domino will topple if the kth domino topples, then eventually all will topple.

DEMO

SUMS OF SERIES Example 2 Prove that

n X

i2 =

i=1

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Find 12 + 22 + 32 + 42 + ::::: + 1002 :

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n(n + 1)(2n + 1) for all n 2 Z + . 6

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n(n + 1)(2n + 1) ” for all n 2 Z + . 6 Proof: (By the principle of mathematical induction)

Pn is: “12 + 22 + 32 + 42 + ::::: + n2 =

(1)

LHS = 12 = 1 and RHS =

If n = 1, )

(2)

P1 is true

1£2£3 =1 6

If Pk is true, then k X

i2 = 12 + 22 + 32 + 42 + ::::: + k 2 =

i=1

k(k + 1)(2k + 1) 6

12 + 22 + 32 + 42 + ::::: + k2 + (k + 1)2

Thus

k(k + 1)(2k + 1) + (k + 1)2 fusing Pk g 6 6 k(k + 1)(2k + 1) = + (k + 1)2 £ Always look 6 6 for common (k + 1)[k(2k + 1) + 6(k + 1)] factors. = 6 (k + 1)(2k2 + k + 6k + 6) = 6

=

(k + 1)(2k2 + 7k + 6) 6 (k + 1)(k + 2)(2k + 3) = 6 (k + 1)([k + 1] + 1)(2[k + 1] + 1) = 6 =

Thus Pk+1 is true whenever Pk is true. Since P1 is true, fPrinciple of mathematical inductiong Pn is true for all n 2 Z + : b

12 + 22 + 32 + 42 + ::::: + 1002 =

100 £ 101 £ 201 = 338 350 fas n = 100g 6

EXERCISE 9B 1 Use the principle of mathematical induction to prove that the following propositions (conjectures) are true for all positive integers n: n n X X n(n + 1) n(n + 1)(n + 2) i= i(i + 1) = b a 2 3 i=1 i=1 n X

c

3i(i + 4) =

i=1

n(n + 1)(2n + 13) 2

d

n X i=1

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Note: You should remember the result from a: n(n + 1) 1 + 2 + 3 + :::::: + n = 2

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2 Prove that the following proposition is true for all positive integers n: n X i £ 2i¡1 = (n ¡ 1) £ 2n + 1 i=1

Example 3 Prove that: n X 1 n = (3i ¡ 1)(3i + 2) 6n + 4 i=1 Pn is: “

for all n 2 Z + :

1 1 1 n + + :::::: + = ” for all n 2 Z + . 2£5 5£8 (3n ¡ 1)(3n + 2) 6n + 4

Proof: (By the principle of mathematical induction) 1 1 1 and RHS = = 10 = (1) If n = 1, LHS = 2£5 6£1+4 ) P1 is true.

1 10

(2) If Pk is true, then k X i=1

1 1 1 1 k = + + :::::: + = (3i ¡ 1)(3i + 2) 2£5 5£8 (3k ¡ 1)(3k + 2) 6k + 4 1 1 1 1 + + :::::: + + 2£5 5£8 (3k ¡ 1)(3k + 2) (3k + 2)(3k + 5)

Now =

k 1 + 6k + 4 (3k + 2)(3k + 5)

fusing Pk g

k 1 + 2(3k + 2) (3k + 2)(3k + 5) µ ¶ µ ¶ 3k + 5 2 k 1 = £ + £ 2(3k + 2) 3k + 5 (3k + 2)(3k + 5) 2 =

=

3k2 + 5k + 2 2(3k + 2)(3k + 5)

=

(3k + 2)(k + 1) 2(3k + 2)(3k + 5)

=

k+1 6k + 10

=

[k + 1] 6[k + 1] + 4

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Thus P1 is true, and Pk+1 is true whenever Pk is true. fPrinciple of mathematical inductiong ) Pn is true for all n 2 Z + :

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3 Prove that the following propositions are true for n 2 Z + : 1 1 1 1 n a + + + ::::: + = 1£2 2£3 3£4 n(n + 1) n+1 1 1 1 1 and hence find + + + ::::: + 10 £ 11 11 £ 12 12 £ 13 20 £ 21 1 1 1 n(n + 3) b + + ::::: + = 1£2£3 2£3£4 n(n + 1)(n + 2) 4(n + 1)(n + 2) 4 Prove that the following propositions are true for n 2 Z + : a 1 £ 1! + 2 £ 2! + 3 £ 3! + 4 £ 4! + ::::: + n £ n! = (n + 1)! ¡ 1 where n! is the product of the first n positive integers. 1 2 3 4 n (n + 1)! ¡ 1 + + + + ::::: + = , and hence find the sum 2! 3! 4! 5! (n + 1)! (n + 1)! 1 2 3 4 9 + + + + ::::: + in rational form. 2! 3! 4! 5! 10!

b

5 Prove that the following conjecture is true: n + 2(n ¡ 1) + 3(n ¡ 2) + ::::: + (n ¡ 2)3 + (n ¡ 1)2 + n = for all integers n > 1:

n(n + 1)(n + 2) 6

1£6+2£5+3£4+4£3+5£2+6£1 = 1 £ 5 + 2 £ 4 + 3 £ 3 + 4 £ 2 + 5 £ 1 + (1 + 2 + 3 + 4 + 5 + 6)

Hint:

DIVISIBILITY Consider the expression 4n + 2 for n = 0, 1, 2, 3, 4, 5, ..... 40 + 2 = 3 = 3 £ 1 41 + 2 = 6 = 3 £ 2 42 + 2 = 18 = 3 £ 6 43 + 2 = 66 = 3 £ 22 44 + 2 = 258 = 3 £ 86

We observe that each of the answers is divisible by 3 and so we make the conjecture “4n + 2 is divisible by 3 for all n 2 Z + .”

This proposition may or may not be true. If it is true, then we should be able to prove it using the principle of mathematical induction. 4n + 2 can be proven to be divisible by 3 by using the binomial expansion. We observe that

Note:

4n + 2 = (1 + 3)n + 2 ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ n ¢ n¡1 ¡n¢ n = 1n + n1 3 + n2 32 + n3 33 + n4 34 + :::::: + n¡1 3 + n 3 +2 ¡n¢ ¡n¢ 2 ¡n¢ 3 ¡n¢ 4 ¡ n ¢ n¡1 ¡n¢ n = 3 + 1 3 + 2 3 + 3 3 + 4 3 + :::::: + n¡1 3 + n 3 ³ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ n ¢ n¡2 ¡n¢ n¡1 ´ 3 = 3 1 + n1 + n2 3 + n3 32 + n4 33 + :::::: + n¡1 + n 3

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where the contents of the brackets is an integer.

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Example 4 Prove that 4n + 2 is divisible by 3 for n 2 Z , n > 0. Pn is: “4n + 2 is divisible by 3” for n 2 Z , n > 0. Proof: (By the principle of mathematical induction) ) P0 is true. (1) If n = 0, 40 + 2 = 3 = 1 £ 3 If Pk is true, then 4k + 2 = 3A where A is an integer Now 4k+1 + 2 = 41 4k + 2 = 4(3A ¡ 2) + 2 fusing Pk g = 12A ¡ 8 + 2 = 12A ¡ 6 = 3(4A ¡ 2) where 4A ¡ 2 is an integer as A 2 Z .

(2)

Thus 4k+1 + 2 is divisible by 3 if 4k + 2 is divisible by 3. Hence P0 is true and Pk+1 is true whenever Pk is true. ) Pn is true for all n 2 Z , n > 0 fPrinciple of mathematical inductiong

6 Use the principle of mathematical induction to prove that: a b c d

n3 + 2n is divisible by 3 for all positive integers n n(n2 + 5) is divisible by 6 for all integers n 2 Z + 6n ¡ 1 is divisible by 5 for all integers n > 0 7n ¡ 4n ¡ 3n is divisible by 12 for all n 2 Z + .

SEQUENCES Example 5 A sequence is defined by: u1 = 1 and un+1 = 2un + 1 for all n 2 Z + : Prove that un = 2n ¡ 1 for all n 2 Z + : Pn is: “if u1 = 1 and un+1 = 2un + 1 for all n 2 Z + , then un = 2n ¡ 1.” Proof: (By the principle of mathematical induction) (1) If n = 1,

u1 = 21 ¡ 1 = 2 ¡ 1 = 1 which is true and so P1 is true.

(2) If Pk is true, then uk = 2k ¡ 1 and uk+1 = 2uk + 1 = 2(2k ¡ 1) + 1 fusing Pk g = 2k+1 ¡ 2 + 1 = 2k+1 ¡ 1 ) Pk+1 is also true.

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Thus, P1 is true and Pk+1 is true whenever Pk is true, so Pn is true for all n 2 Z + fPrinciple of mathematical inductiong

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7 Use the principle of mathematical induction to prove these propositions: a If sequence fun g is defined by: u1 = 5 and un+1 = un + 8n + 5 for all n 2 Z + , then un = 4n2 + n. b If the first term of a sequence is 1 and subsequent terms are defined by the recursion formula un+1 = 2 + 3un , then un = 2(3n¡1 ) ¡ 1. un c If a sequence is defined by: u1 = 2 and un+1 = for all n 2 Z + , 2(n + 1) 22¡n then un = . n! d If a sequence is defined by: u1 = 1 and un+1 = un + (¡1)n (n + 1)2 for all (¡1)n¡1 n(n + 1) n 2 Z + , then un = . 2 8 A sequence is defined by u1 = 1 and un+1 = un + (2n + 1) for all n 2 Z + . By finding un for n = 2, 3 and 4, conjecture a formula for un in terms of n only. Prove that your conjecture is true using the principle of mathematical induction. 1 for all n 2 Z + . (2n + 1)(2n + 3) By finding un for n = 2, 3 and 4, conjecture a formula for un in terms of n only. Prove that your conjecture is true using the principle of mathematical induction. p p 10 (2 + 3)n = An + Bn 3 for all n 2 Z + , where An and Bn are integers. 9 A sequence is defined by u1 =

a b c d

1 3

and un+1 = un +

Find An and Bn for n = 1, 2, 3 and 4. Without using induction, show that An+1 = 2An + 3Bn and Bn+1 = An + 2Bn . Calculate (An )2 ¡ 3(Bn )2 for n = 1, 2, 3 and 4 and hence conjecture a result. Prove that your conjecture is true.

11 Prove that un =

2n ¡ (¡1)n 3

is an odd number for all n 2 Z + .

12 Another form of the principle of mathematical induction is: If Pn is a proposition defined for all n 2 Z + , and if (1) P1 and P2 are true, and (2) Pk+2 is true whenever Pk and Pk+1 are true, then Pn is true for all n 2 Z + . Use this form to prove that:

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a If a sequence un is defined by u1 = 11, u2 = 37 and un+2 = 5un+1 ¡ 6un for all n 2 Z + , then un = 5(3n ) ¡ 2n+1 . p p b If un = (3 + 5)n + (3 ¡ 5)n where n 2 Z + , then un is a multiple of 2n : Hint: First find a and b such that un+2 = aun+1 + bun

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OTHER APPLICATIONS Proof by the principle of mathematical induction is used in several other areas of mathematics. For example, in establishing truths dealing with: ² ² ²

² ² ²

inequalities differential calculus matrices

geometrical generalisations products complex numbers.

You will find some proofs with these topics in the appropriate chapters later in the book.

Example 6 Prove that a convex n-sided polygon has

1 2 n(n

¡ 3) diagonals for all n > 3.

Pn is: “A convex n-sided polygon has

1 2 n(n

¡ 3) diagonals for all n > 3”.

Proof: (By the principle of mathematical induction) (1) If n = 3 we have a triangle and 12 £ 0 £ (¡3) = 0

There are 0 diagonals.

) P3 is true. (2)

If Pk is true, a convex k-sided polygon has If we label the vertices 5 1, 2, 3, 4, 5, ....., k ¡ 1, k 4 and k + 1 as an additional vertex 3 then

1 2 k(k

¡ 3) diagonals. k-4 k-3 k-2

2

k-1 1

k k+1

¡ 2} +1 Pk+1 = Pk + k | {z

the line from 1 to k was once a side and is now a diagonal

the number of diagonals from k + 1 to the vertices 2, 3, 4, 5, ....., k ¡ 1 Try it for k¡=3 ¡ ,¡4,.... ¡

Pk+1 = 12 k(k ¡ 3) + k ¡ 1

)

= 12 k(k ¡ 3) + 22 (k ¡ 1) = 12 [k2 ¡ 3k + 2k ¡ 2] = 12 [k2 ¡ k ¡ 2] = 12 (k + 1)(k ¡ 2) = 12 (k + 1)([k + 1] ¡ 3)

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Thus P3 is true and Pk+1 is true whenever Pk is true. ) Pn is true for all n > 3 fPrinciple of mathematical inductiong

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13 Use the principle of mathematical induction to prove the following propositions: µ ¶µ ¶µ ¶µ ¶ µ ¶ 1 1 1 1 1 1 a 1¡ 1¡ 1¡ 1¡ ::::: 1 ¡ = , n 2 Z +. 2 3 4 5 n+1 n+1 ¶ n µ X 1 Note: The LHS 6= 1¡ since it is a product not a sum. i+1 i=1 b If n straight lines are drawn on a plane such that each line intersects every other line and no three lines have a common point of intersection, then the plane is divided n(n + 1) into + 1 regions. 2 c If n points are placed inside a triangle and non-intersecting lines are drawn connecting the 3 vertices of the triangle and the points within it to partition the triangle into smaller triangles, then the number of triangles resulting is 2n + 1. µ ¶µ ¶µ ¶ µ ¶ 1 1 1 1 n+1 d 1¡ 2 1¡ 2 1 ¡ 2 ::::: 1 ¡ 2 = for all integers n > 2: 2 3 4 n 2n 14 Prove the following propositions to be true using the principle of mathematical induction: a 3n > 1 + 2n for all n 2 Z , n > 0 b n! > 2n

for all n 2 Z , n > 4

3

for all n 2 Z +

c 8 >n n

d (1 ¡ h)n 6

1 1 + nh

for 0 6 h 6 1 and all n 2 Z + .

INVESTIGATION

SEQUENCES, SERIES AND INDUCTION

This investigation involves the principle of mathematical induction as well as concepts from sequences, series, and counting. What to do:

1 The sequence of numbers fun g is defined by u1 = 1 £ 1!, u2 = 2 £ 2!, u3 = 3 £ 3!, etc. What is the nth term of the sequence? 2 Let Sn = u1 + u2 + u3 + ::::: + un . Investigate Sn for several different values of n. 3 Based on your results from 2, conjecture an expression for Sn . 4 Prove your conjecture to be true using the principle of mathematical induction. 5 Show that un can be written as (n + 1)! ¡ n! and devise an alternative direct proof of your conjecture for Sn . 6 Let Cn = un + un+1 . Write an expression for Cn in factorial notation and simplify it. 7 Let Tn = C1 + C2 + C3 + ::::: + Cn

and find Tn for n = 1, 2, 3, 4 and 5.

8 Conjecture an expression for Tn .

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9 Prove your conjecture for Tn by any method.

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INDIRECT PROOF (EXTENSION)

C

Some propositions may be proven to be true by using an indirect proof such as proof by contradiction. In such proofs we suppose the opposite of the statement to be true and, by using correct argument, hope to obtain a contradiction.

Example 7 Prove that the sum of any positive real number and its reciprocal is at least 2. Proof: (by contradiction) 1 Suppose that x + < 2 for some x > 0 x µ ¶ 1 ) x x+ < 2x fmultiplying both sides by x where x > 0g x ) x2 + 1 < 2x ) x2 ¡ 2x + 1 < 0 ) (x ¡ 1)2 < 0 which is a contradiction as no perfect square of a real number can be negative. 1 So, the supposition is false and its opposite x + > 2, x > 0 must be true. x

Example 8 Prove that the solution of 2x = 3 is an irrational number. Proof: (by contradiction) Suppose that if 2x = 3 then x is rational )

p

2 q = 3 for some positive integers p and q, q 6= 0 p

)

(2 q )q = 3q ) 2p = 3q which is clearly a contradiction, as the LHS = 2p is even and the RHS = 3q is odd. ) the supposition is false and its opposite is true i.e., if 2x = 3 then x is irrational.

p where p and q are q integers, q = 6 0 and p, q have no common factors.

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A rational number can be written in the form

Reminder:

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EXERCISE 9C 1 Use proof by contradiction to prove that: a the sum of a positive number and nine times its reciprocal is at least 6 b the solution of 3x = 4 is an irrational number c log2 5 is irrational. p 2 Challenge: Prove by contradiction that 2 is irrational.

REVIEW SET 9A Prove the following propositions using the principle of mathematical induction: n X (2i ¡ 1) = n2 , n 2 Z + . 1 i=1

2 7n + 2 is divisible by 3, n 2 Z + . 3

n X

i(i + 1)(i + 2) =

i=1

n(n + 1)(n + 2)(n + 3) , n 2 Z +. 4

4 1 + r + r2 + r3 + r4 + ::::: + rn¡1 =

1 ¡ rn , n 2 Z + , provided that r 6= 1. 1¡r

5 52n ¡ 1 is divisible by 24, n 2 Z + : 6 5n > 1 + 4n, n 2 Z + : 7 If u1 = 1 and un+1 = 3un + 2n , then un = 3n ¡ 2n , n 2 Z + .

REVIEW SET 9B Prove the following propositions using the principle of mathematical induction: 1

n X

(2i ¡ 1)2 =

i=1

n(2n + 1)(2n ¡ 1) 3

n 2 Z + , n > 1:

2 32n+2 ¡ 8n ¡ 9 is divisible by 64 for all positive integers n. 3

n X

(2i + 1)2i¡1 = 1 + (2n ¡ 1) £ 2n

for all positive integers n.

i=1

4 5n + 3 is divisible by 4 for all integers n > 0: 5

n X

i(i + 1)2 =

i=1

n(n + 1)(n + 2)(3n + 5) 12

for all positive integers n.

6 5n + 3n > 22n+1 , n 2 Z + :

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7 If u1 = 9 and un+1 = 2un + 3(5n ), then un = 2n+1 + 5n , n 2 Z + .

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REVIEW SET 9C Prove the following propositions using the principle of mathematical induction: 1

n X

i(i + 2) =

i=1

n(n + 1)(2n + 7) , n 2 Z +: 6

2 7n ¡ 1 is divisible by 6, n 2 Z + : 3

n X

(2i ¡ 1)3 = n2 (2n2 ¡ 1) for all positive integers n > 1:

i=1

4 3n ¡ 1 ¡ 2n is divisible by 4 for all non-negative integers n. 5

n X i=1

1 n = for all positive integers n. (2i ¡ 1)(2i + 1) 2n + 1

6 If u1 = 5 and un+1 = 2un ¡ 3n(¡1)n , then un = 3(2n ) + (¡1)n n, n 2 Z + .

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Chapter

10

The unit circle and radian measure Contents:

A B C D

Radian measure Arc length and sector area The unit circle and the basic trigonometric ratios Areas of triangles

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THE UNIT CIRCLE AND RADIAN MEASURE (Chapter 10)

Before starting this chapter you should make sure that you have a good understanding of the necessary background knowledge in trigonometry and Pythagoras. Click on the icon alongside to obtain a printable set of exercises and answers on this background knowledge.

BACKGROUND KNOWLEDGE

OPENING PROBLEM Consider an equilateral triangle with sides 10 cm long. All its angles are of size 60o . Altitude AN bisects side BC and the vertical angle BAC. Can you see from this figure that sin 30o = 12 ? Use your calculator to find the values of sin 30o , sin 150o , sin 390o , sin 1110o and sin(¡330o ). What do you notice? Can you explain why this result occurs even though the angles are not between 0o and 90o ?

² ²

A

A

30° 30°

60° B

N

10 cm

60° C 5 cm

RADIAN MEASURE

DEGREE MEASUREMENT OF ANGLES Recall that one full revolution makes an angle of 360o and a straight angle is 180o . Hence, 1 th of one full revolution. This measure of angle is one degree, 1o , can be defined as 360 probably most useful for surveyors and architects, and is the one you have probably used in earlier years. 1 th of one degree and one second, 100 , For greater accuracy we define one minute, 10 , as 60 1 th of one minute. Obviously a minute and a second are very small angles. as 60

Most graphics calculators have the capacity to convert fractions of angles measured in degrees into minutes and seconds. This is also useful for converting fractions of hours into minutes 1 1 and seconds for time measurement, as one minute is 60 th of one hour, and one second is 60 th of one minute.

RADIAN MEASUREMENT OF ANGLES An angle is said to have a measure of 1 radian (1c ) if it is subtended at the centre of a circle by an arc equal in length to the radius. The symbol ‘c’ is used for radian measure but is usually omitted, whilst the degree symbol is always used r 60° when the measure of an angle is r given in degrees. c

arc length =¡r 1c radius¡=¡r

From the diagram to the right, it can be seen that 1c is slightly smaller than 60o . In fact, 1c ¼ 57:3o .

1

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The word “radian” is an abbreviation of “radial angle”.

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DEGREE-RADIAN CONVERSIONS If the radius of a circle is r, then an arc of length 2r will subtend an angle of 2 radians at the centre. An arc of length ¼r (half the circumference) will subtend an angle of ¼ radians. ¼ radians ´ 180o .

Therefore, So, 1c =

¡ 180 ¢o ¼

¼ 57:3o

and 1o =

¡

¼ 180

¢c

¼ 0:0175c

To convert from degrees to radians, multiply by

¼ 180 :

To convert from radians to degrees, multiply by

180 ¼ :

We can summarise these results in the conversion diagram:

If degrees are used we indicate this with a small o. To indicate radians, we can use a small c or else use no symbol at all.

£

¼ 180

Degrees

Radians

£ 180 ¼

Example 1 Convert 45o to radians in terms of ¼. 45o = (45 £ =

¼ 4

¼ 180 )

radians

180o = ¼ radians ¡ 180 ¢o = ¼4 radians 4

or )

radians

)

45o =

¼ 4

radians

Example 2 Convert 126:5o to radians.

126:5o ¼ = (126:5 £ 180 ) radians ¼ 2:21 radians (3 s.f.)

Example 3 Convert degrees.

5¼ 6

5¼ 6

to =

¡ 5¼ 6

£

¢ 180 o ¼

Notice that angles in radians are expressed either in terms of p or as decimals.

= 150o

Example 4

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0:638 radians o = (0:638 £ 180 ¼ ) o ¼ 36:6

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Convert 0:638 radians to degrees.

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EXERCISE 10A 1 Convert to radians, in b a 90o f 135o g o k 315 l

terms of ¼: 60o c 225o h o 540 m

30o 270o 36o

2 Convert to radians (correct to 3 s.f.): b 137:2o c 317:9o a 36:7o

d

3 Convert the following radian measure to degrees: b 3¼ c 3¼ a ¼5 5 4

d

7¼ 9

f

¼ 10

g

3¼ 20

h

18o 360o 80o

d i n

219:6o ¼ 18 5¼ 6

i

4 Convert the following radians to degrees (to 2 decimal places): a 2 b 1:53 c 0:867 d 3:179

e j o

e

e j

e

9o 720o 230o 396:7o ¼ 9 ¼ 8

5:267

5 Copy and complete: a

Degrees Radians

0

45

90

135 180 225 270 315 360

b

Degrees Radians

0

30

60

90

B

120 150 180 210 240 270 300 330 360

ARC LENGTH AND SECTOR AREA

ARC LENGTH

In the diagram, µ is measured in radians.

A l q

O

arc length µ = circumference 2¼ l µ ) = 2¼r 2¼

B r

) l = µr

AREA OF SECTOR

µ is measured in radians.

X

area of minor sector XOY µ = area of circle 2¼ A µ ) = ¼r2 2¼

r O

q

Y r

) A = 12 µr2

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µ £ ¼r2 . 360

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and A =

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µ £ 2¼r 360

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If µ is in degrees, l =

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Example 5 A sector has radius 12 cm and angle 65o . Use radians to find its: a arc length b area a

area = 12 µr2 65¼ = 12 £ £ 122 180 ¼ 81:7 cm2

b

arc length = µr 65¼ = £ 12 180 ¼ 13:6 cm

Example 6 A sector has radius 8:2 cm and arc length 13:3 cm. Find its angle in radians and degrees. fµ in radiansg

l = µr

13:3 l = ¼ 1:62c r 8:2 13:3 180 and µ = £ ¼ 92:9o 8:2 ¼ )

µ=

EXERCISE 10B 1 Use radians to find: i the arc length a radius 9 cm and angle 41:6o

ii b

the area of a sector of a circle of radius 4:93 cm and angle 122o

2 A sector has an angle of 107:9o and an arc length of 5:92 m. Find: a its radius b its area. 3 A sector has an angle of 68:2o and an area of 20:8 cm2 . Find: a its radius b its perimeter. 4 Find the angle of a sector of: a radius 4:3 m and arc length 2:95 m

radius 10 cm and area 30 cm2 .

b

5 Find µ (in radians) for each of the following, and hence find the area of each figure: a b c 31.7 cm

q

8 cm 6 cm

5 cm

q

8 cm 8.4 cm

q

6 Find the arc length and area of a sector of radius 5 cm and angle 2 radians.

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7 If a sector has radius 10 cm and arc length 13 cm, find its area.

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8 This cone

is made from this sector: 10 cm

s cm q°

12 cm

Find correct to 3 significant figures: a the slant length s cm b c the arc length of the sector d 9 A

the value of r the sector angle (µo )

The end wall of a building has the shape illustrated, where the centre of arc AB is at C. Find: a ® to 4 significant figures b µ to 4 significant figures c the area of the wall.

B

30 m

r cm

5m q° a° C

1 nautical mile (nmi)

10 A nautical mile (nmi) is the distance on the Earth’s surface that subtends an angle of 1 minute (where 1 1 degree) of the Great Circle arc measured minute = 60 from the centre of the Earth. A knot is a speed of 1 nautical mile per hour. a Given that the radius of the Earth is 6370 km, show that 1 nmi is approximately equal to 1:853 km. b Calculate how long it would take a plane to fly from Perth to Adelaide (a distance of 2130 km) if the plane can fly at 480 knots.

N 1'

P Q

C

S

11

A sheep is tethered to a post which is 6 m from a long fence. The length of rope is 9 m. Find the area which is available for the sheep to feed on.

fence 6m post

S

12 A belt fits tightly around two pulleys of radii 4 cm and 6 cm respectively which have a distance of 20 cm between their centres.

a°

f°

q°

Find, correct to 4 significant figures:

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d the length of the belt

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C

THE UNIT CIRCLE AND THE BASIC TRIGONOMETRIC RATIOS y

The unit circle is the circle with centre (0, 0) and radius 1 unit.

1

x

1 the unit circle

CIRCLES WITH CENTRE (0, 0) y

Consider a circle with centre (0, 0) and radius r units, and suppose P(x, y) is any point on this circle.

P(x, y) r

Since OP = r, then p (x ¡ 0)2 + (y ¡ 0)2 = r fdistance formulag

x

(0,¡0)

x2 + y 2 = r2

)

x2 + y2 = r2 is the equation of a circle with centre (0, 0) and radius r. The equation of the unit circle is x2 + y2 = 1.

So,

ANGLE MEASUREMENT

y Positive direction

Suppose P lies anywhere on the unit circle and A is (1, 0). Let µ be the angle measured from [OA] on the positive x-axis. q

µ is positive for anticlockwise rotations and negative for clockwise rotations.

x

1 A

P

For example, µ = 210o

and Á = ¡150o .

y

Negative direction

q x

30°

f

DEFINITION OF SINE, COSINE AND TANGENT So, as point P moves anywhere on the unit circle, cos µ is the x-coordinate of P sin µ is the y-coordinate of P

1

1 -1

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1

P(cos¡q, sin¡q) q

-1

where µ is the angle made by [OP] with the positive x-axis.

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For all points on the unit circle: ¡1 6 x 6 1 and ¡1 6 y 6 1 So, ¡1 6 cos µ 6 1 and ¡1 6 sin µ 6 1 for all µ. Also, we have already seen that the equation of the unit circle is x2 + y 2 = 1. cos2 µ + sin2 µ = 1 for all µ.

This leads to the identity

tan µ =

The tangent ratio is defined as

sin µ cos µ

By considering the first quadrant, we can easily see that the right angled triangle definitions of sine, cosine and tangent are consistent with the unit circle definition, but are restricted to acute angles only. The unit circle definition applies to angles of any value. From the definition, cos ¼2 = 0 and sin ¼2 = 1 If any integer multiple of 2¼ is added to µ, P will still be at (0, 1). ¡ ¢ ¡ ¢ Therefore, cos ¼2 + 2k¼ = 0 and sin ¼2 + 2k¼ = 1, k 2 Z : More generally, adding integer multiples of 2¼ to any value of µ will not change the position of P. So for all k 2 Z and angles µ, cos (µ + 2k¼) = cos µ and sin (µ + 2k¼) = sin µ. This periodic feature is an important property of the trigonometric functions.

Example 7 Use the unit circle to show that cos

¡¼ 2

¢ + µ = ¡ sin µ.

Consider point B(a, b) on the unit circle such that [OB] makes an angle of µ with the positive x-axis. Thus a = cos µ and b = sin µ. ¢ ¡ Now consider point P on the unit circle such that [OP] makes an angle of ¼2 + µ with the positive x-axis. y b = ¼ Now OB = OP = 1 unit and BOP 2 P so P is obtained from B by an anticlockwise B(a,¡b) ¼ o rotation of 2 or 90 about the origin. b ) from transformation geometry, P has q a coordinates (¡b, a) or (¡ sin µ, cos µ). x But the coordinates of P are (cos( ¼2 + µ), sin( ¼2 + µ)). ) cos( ¼2 + µ) = ¡ sin µ

Example 8

y (0'\1)

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) cos(¡270o ) = 0 and sin(¡270o ) = 1

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x -270°

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Use a unit circle diagram to find the values of cos(¡270o ) and sin(¡270o ):

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Example 9 Find the possible values of cos µ for sin µ = 23 . Illustrate. cos2 µ + sin2 µ = 1 ¡ ¢2 ) cos2 µ + 23 = 1 cos2 µ =

) )

cos µ =

1 y We_ 1

5 9

p § 35

~`5 -\___ 3

x

~`5 ___ 3

Helpful hint: We have seen previously how the quadrants of the Cartesian Plane are labelled in anticlockwise order from 1st to 4th. We can also use a letter to show which trigonometric ratios are positive in each quadrant. You might like to remember them using

y

2nd

1st

S

A

T

C

3rd

All Silly Turtles Crawl.

x 4th

Example 10 If sin µ = ¡ 34 and ¼ < µ
0

¡ p12 ; p12

(1, 0)

7p 4

³

´

100

Hence,

a=

³

50

)

1 2 p1 2

75

a2 =

)

fPythagorasg

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´

(0,-1)

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THE UNIT CIRCLE AND RADIAN MEASURE (Chapter 10) ¼ 6

MULTIPLES OF ¼ 3

Consider µ =

Triangle OAP is isosceles with vertical angle 60o. The remaining angles are therefore 60o and so triangle AOP is equilateral. The altitude [PN] bisects base [OA],

= 60o :

y 1

P(1–2 , k)

1

) ON =

If P is ( 12 , k), then ( 12 )2 + k2 = 1 ) k2 = 34

k

60°

x

A(1'\\0)

Qw_ N

1 2

) ) P is ( 12 , Consequently, we can find the coordinates of all points on the unit circle corresponding to multiples of ¼6 using rotations and reflections.

p 3 2

fas k > 0g

p

3 2 )

´ ³ p ¡ 23 ; 12

p 3 2

where y (0,¡1)

³ p ´ ¡ 12 ; 23

So we have:

k=

³

p ´ 1 3 2; 2

³p 7p 6

3 1 2 ; 2

2p 3

¡ 12 ; ¡

p

3 2

´

x (1,¡0) ´ ³p 3 1 2 ; ¡2

(-1,¡0) ³ p ´ ¡ 23 ; ¡ 12

³

¼ 0:9

´

³ (0,-1)

p ´ 1 3 2;¡ 2

Summary: ²

If µ is a multiple of and §1.

¼ , 2

the coordinates of the points on the unit circle involve 0

²

If µ is a multiple of

¼ , 4

(but not a multiple of

¼ 2 ),

the coordinates involve § p12 :

²

If µ is a multiple of

¼ , 6

(but not a multiple of

¼ 2 ),

the coordinates involve § 12

and

p § 23 .

You should not try to memorise the coordinates on the above circles for every multiple of and ¼4 , but rather use the summary to work out the correct result. Consider 225o =

For example:

5¼ 4

y

y 5p 4

5p 3

x

¡ p12 ; ¡ p12

´ »-0.9

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x ³

p ´ 1 3 2;¡ 2

is in quad 4, so signs are (+, ¡) and from the diagram the x-value is 12 .

is in quad 3, so signs are both negative and both have p12 size.

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Consider 300o = i.e., a multiple

³

¼ 6

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Example 15 Use a unit circle to find the exact values of sin ®, cos ® and tan ® for ® = ³

¡ p12 ; p12

3¼ 4 :

y

´

» 0.7 3p 4

) x

3¼ p1 cos( 3¼ 4 ) = ¡ 2 , sin( 4 ) = ¡ ¢ = ¡1 tan 3¼ 4

p1 2

»-0.7

Example 16 y

Use a unit circle diagram to find the exact values of sin A, cos A and tan A for A = 4¼ 3 :

x

p ´ ¡ 3 2

1 2;

p 3 2 p ¡ 23 ¡ 12

and sin( 4¼ 3 ) = ¡

4p 3

-1–2

³

1 ) cos( 4¼ 3 ) = ¡2

) tan( 4¼ 3 ) =

»-0.9

) tan( 4¼ 3 ) =

p 3

Example 17 Without using a calculator, find the value of 8 sin( ¼3 ) cos( 5¼ 6 ): ³

y

³

´ p ¡ 3 1 2 ;2

p ´ 1 3 2; 2

sin( ¼3 ) =

5p 6

)

x

p 3

p 3 2

p 3 2 p p 8( 23 )(¡ 23 )

and cos( 5¼ 6 )=¡

8 sin( ¼3 ) cos( 5¼ 6 ) =

= 2(¡3) = ¡6

Example 18 Use a unit circle diagram to find all angles in [0, 2¼] with a cosine of 12 . As the cosine is 12 , we draw the vertical line x = 12 :

y 5p 3

p 3

Because 12 is involved we know the required angles are multiples of ¼6 . They are ¼3 and 5¼ 3 .

x Qw_

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EXERCISE 10C.3 1 Use a unit circle diagram to find sin µ, cos µ and tan µ for µ equal to: b 5¼ c 7¼ d ¼ e a ¼4 4 4

¡3¼ 4

2 Use a unit circle diagram to find sin ¯, cos ¯ and tan ¯ for ¯ equal to: ¼ 6

a

2¼ 3

b

7¼ 6

c

d

5¼ 3

11¼ 6

e

3 Without using a calculator, evaluate: a

sin2 60o

d

1 ¡ cos2

g j

c

4 sin 60o cos 30o

e

sin 30o cos 60o ¡ ¢ sin2 2¼ 3 ¡1

f

cos2 ( ¼4 ) ¡ sin( 7¼ 6 )

5¼ sin( 3¼ 4 ) ¡ cos( 4 )

h

1 ¡ 2 sin2 ( 7¼ 6 )

i

2 5¼ cos2 ( 5¼ 6 ) ¡ sin ( 6 )

tan2 ( ¼3 ) ¡ 2 sin2 ( ¼4 )

k

3¼ 2 tan(¡ 5¼ 4 ) ¡ sin( 2 )

l

2 tan 150o 1 ¡ tan2 150o

b

¡¼¢ 6

Check all answers using your calculator. 4 Use a unit circle diagram to find all angles between 0o and 360o with: p

1 2

a

a sine of

d

a cosine of ¡ 12

3 2

b

a sine of

e

a cosine of ¡ p12

p1 2 p 3 ¡ 2

c

a cosine of

f

a sine of

5 Use a unit circle diagram to find all angles between 0 and 2¼ which have: p a a tangent of 1 b a tangent of ¡1 c a tangent of 3 p d a tangent of 0 e a tangent of p13 f a tangent of ¡ 3 6 Use a unit circle diagram to find all angles between 0 and 4¼ with: a

a cosine of

p 3 2

b

a sine of ¡ 12

7 Find µ in radians if 0 6 µ 6 2¼

and:

1 2

3 2

a

cos µ =

e i

p

b

sin µ =

cos µ = ¡ p12

f

sin2 µ = 1

tan µ = ¡ p13

j

tan2 µ = 3

D

c

a sine of ¡1

c

cos µ = ¡1

d

sin µ = 1

g

cos2 µ = 1

h

cos2 µ =

1 2

AREAS OF TRIANGLES DEMO

height base

base

base

If we know the base and height measurements of a triangle we can calculate the area using

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However, cases arise where we do not know the height but we can still calculate the area. These cases are: ² knowing two sides and the included angle between them For example:

² knowing all three sides

8 cm

8 cm

9 cm

49° 10 cm

10 cm

USING THE INCLUDED ANGLE If triangle ABC has angles of size A, B and C, the sides opposite these angles are labelled a, b and c respectively. Using trigonometry, we can develop an alternative formula that does not depend on a perpendicular height.

A A c B

B

Any triangle that is not right angled must be either acute or obtuse. We will consider both cases:

b C a

A

A

c

c

b

h

B

C

D

h

b B

C

a

a

(180°-C)

D

C

In both triangles a perpendicular is constructed from A to D on [BC] (extended if necessary). h b h = b sin C

)

So, area = 12 ah gives

h b ) h = b sin(180o ¡ C) but sin(180o ¡ C) = sin C ) h = b sin C sin(180o ¡ C) =

sin C =

area = 12 ab sin C.

1 2 bc sin A

Using different altitudes we can show that the area is also

or

1 2 ac sin B:

Summary: Given the lengths of two sides of a triangle and the included angle between them, the area of the triangle is a half of the product of two sides and the sine of the included angle.

side included angle

side

Example 19 Area = 12 ac sin B

Find the area of triangle ABC: A

=

11 cm

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£ 15 £ 11 £ sin 28o

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28°

B

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Example 20 A triangle has sides of length 10 cm and 11 cm and an area of 50 cm2 . Show that the included angle may have two possible sizes. 1 2

If the included angle measures µo , then Now arcsin

¡ 50 ¢ 55

£ 10 £ 11 £ sin µ = 50 ) sin µ = 50 55

¼ 65:4 10

) µ ¼ 65:4 or 180 ¡ 65:4 i.e., µ ¼ 65:4 or 114:6

10 11

65.4°

11

114.6°

So, the two different possible angles are 65:4o and 114:6o .

EXERCISE 10D 1 Find the area of: a

b

9 cm

c

31 km 82°

40°

10.2 cm

25 km

2p 3

10 cm

6.4 cm

A

2

2 If triangle ABC has area 150 cm , find the value of x:

17 cm B

68° C

x cm

3 A parallelogram has two adjacent sides of length 4 cm and 6 cm respectively. If the included angle measures 52o , find the area of the parallelogram. 4 A rhombus has sides of length 12 cm and an angle of 72o . Find its area. 5 Find the area of a regular hexagon with sides of length 12 cm. 6 A rhombus has an area of 50 cm2 and an internal angle of size 63o . Find the length of its sides. 7 A regular pentagonal garden plot has centre of symmetry O and an area of 338 m2 . Find the distance OA.

A

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8 Find the possible values of the included angle of a triangle with: a sides 5 cm and 8 cm, and area 15 cm2 b sides 45 km and 53 km, and area 800 km2 .

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9 The Australian 50 cent coin has the shape of a regular dodecagon (12 sides). Eight of these 50 cent coins will fit exactly on an Australian $10 note as shown. What fraction of the $10 note is not covered? 10 Find the shaded area in: a

b A 12 cm

12 cm

O

B 18 cm P 0.66 c

1.5

HERON’S FORMULA In the first century A.D., Heron of Alexandria showed that if a triangle has sides of length a, b and c, then its area can be calculated using A=

p s(s ¡ a)(s ¡ b)(s ¡ c)

s=

where

a+b+c . 2

a Find the area of the right angled triangle with sides 3 cm, 4 cm and 5 cm: i without using Heron’s formula ii using Heron’s formula. b Find the area of a triangle with sides of length: i 6 cm, 8 cm and 12 cm ii 7:2 cm, 8:9 cm and 9:7 cm.

11

REVIEW SET 10A 1 Determine the area of:

7.3 km 38° 9.4 km

2 Determine the area of: a a sector of angle 80o and radius 13 cm b a triangle with sides 11 cm, 9 cm and included angle 65o .

y 10°

3 Find the coordinates of the points M, N and P on the unit circle.

N

1 M 73° -53°

1 x

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4 Find the angle [OA] makes with the positive x-axis if the x-coordinate of the point A on the unit circle is ¡0:222 . 5 Find the acute angles that would have the same: b sine as 165o a sine as 2¼ 3

c

cosine as 276o

6 Find the obtuse angles which have the same: ¼ b sine as 15 a sine as 47o

c

cosine as 186o

7 Without using your calculator, find: a sin 159o if sin 21o ¼ 0:358 c cos 75o if cos 105o ¼ ¡0:259

b d

cos 92o if cos 88o ¼ 0:035 sin 227o if sin 47o ¼ 0:731

8 Use a unit circle diagram to find: a cos 360o and sin 360o

b

cos(¡¼) and sin(¡¼)

9 Explain how to use the unit circle to find µ when cos µ = ¡ sin µ. 10 If sin 74o ¼ 0:961, find without using a calculator the value of: b sin 254o c sin 286o d sin 646o a sin 106o ¡ ¢ ¡ ¢ ¡ 3¼ ¢ b cos 3¼ 11 Without a calculator, evaluate: a tan2 2¼ 3 4 ¡ sin 4 12 If sin x = ¡ 14

and ¼ < x
180o

which is impossible.

EXERCISE 11B.2 1 Triangle ABC has angle B = 40o , b = 8 cm and c = 11 cm. Find the two possible values for angle C. 2 In triangle ABC, find the measure of: b = 65o a angle A if a = 14:6 cm, b = 17:4 cm and ABC b b angle B if b = 43:8 cm, c = 31:4 cm and ACB = 43o b = 71o . c angle C if a = 6:5 km, c = 4:8 km and BAC 3 Is it possible to have a triangle with measurements as shown? Explain!

4 Find the magnitude of the angle ABC and hence the length BD.

B

85° 12°

9.8 cm 68°

20 cm

C

A

11.4 cm

5 Find x and y in the given figure.

D 78°

ym

95° xm

30° 118°

22 m

b = 58o , AB = 10 cm and AC = 5:1 cm. Find: 6 Triangle ABC has A b correct to the nearest tenth of a degree using the sine rule a C b correct to the nearest tenth of a degree using the cosine rule. b C

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c Copy and complete: “When faced with using either the sine rule or the cosine rule it is better to use the ..................... as it avoids ...................”

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NON-RIGHT ANGLED TRIANGLE TRIGONOMETRY (Chapter 11)

b = 30o , AC = 9 cm and AB = 7 cm. Find the area of the 7 In triangle ABC, ABC triangle. 8 In the diagram alongside, find the exact value of x. Express your answer in the p form a + b 2 where a, b 2 Q .

C

45° 30°

(x¡+¡3) cm

(2x¡-¡5) cm

USING THE SINE AND COSINE RULES

If we are given a problem involving a triangle, we must first decide which rule to use. If the triangle is right angled then the trigonometric ratios or Pythagoras’ Theorem can be used. For some problems we can add an extra line or two to the diagram to create a right angled triangle. However, if we do not have a right angled triangle and we have to choose between the sine and cosine rules, the following checklist may be helpful: ² ²

Use the cosine rule when given:

three sides two sides and an included angle.

Use the sine rule when given: ² one side and two angles ² two sides and a non-included angle, but beware of the ambiguous case which can occur when the smaller of the two given sides is opposite the given angle.

Example 6 T

The angles of elevation to the top of a mountain are measured from two beacons A and B at sea. These angles are as shown on the diagram. If the beacons are 1473 m apart, how high is the mountain?

A

29.7° 41.2° B

b = 41:2o ¡ 29:7o ATB = 11:5o T

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29.7° 41.2° A 1473 m B

x 1473 = o sin 29:7 sin 11:5o 1473 ) x= £ sin 29:7o sin 11:5o ¼ 3660:62

hm

75

xm

fexterior angle of ¢g

We find x in ¢ABT using the sine rule:

25

11.5°

N

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Now, in ¢BNT, sin 41:2o =

h h ¼ x 3660:62

h ¼ sin 41:2o £ 3660:62 h ¼ 2410

) )

So, the mountain is about 2410 m high.

Example 7

S

R

Find the measure of angle RPV. P

W 5 cm

T U

p p 52 + 32 = 34 cm. p p PV = 62 + 32 = 45 cm. p p PR = 62 + 52 = 61 cm.

V

6 cm

In ¢RVW, RV =

fPythagorasg

In ¢PUV,

fPythagorasg

In ¢PQR,

~`6`1 cm

P

fPythagorasg p p p ( 61)2 + ( 45)2 ¡ ( 34)2 p p cos µ = 2 61 45

R

q

=

~`4`5 cm

3 cm

Q

~`3`4 cm V

)

61 + 45 ¡ 34 p p 2 61 45

72 = p p 2 61 45 ¶ µ 36 µ = cos¡1 p p ¼ 46:6o 61 45

) angle RPV measures about 46:6o .

EXERCISE 11C 1 Rodrigo wishes to determine the height of a flagpole. He takes a sighting of the top of the flagpole from point P. He then moves further away from the flagpole by 20 metres to point Q and takes a second sighting. The information is shown in the diagram alongside. How high is the flagpole?

63 m

112°

P

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P

53°

R

lake

75

175 m

28° 20 m

To get from P to R, a park ranger had to walk along a path to Q and then to R as shown. What is the distance in a straight line from P to R?

Q

100

2

Q

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3 A golfer played his tee shot a distance of 220 m to a point A. He then played a 165 m six iron to the green. If the distance from tee to green is 340 m, determine the number of degrees the golfer was off line with his tee shot.

279

A 165 m G 220 m 340 m T

4

A Communications Tower is constructed on top of a building as shown. Find the height of the tower.

tower 15.9°

building

23.6° 200 m

5 A football goal is 5 metres wide. When a player is 26 metres from one goal post and 23 metres from the other, he shoots for goal. What is the angle of view of the goals that the player sees?

angle of view

goal posts

player

6 A tower 42 metres high stands on top of a hill. From a point some distance from the base of the hill, the angle of elevation to the top of the tower is 13:2o and the angle of elevation to the bottom of the tower is 8:3o . Find the height of the hill. 7 From the foot of a building I have to look upwards at an angle of 22o to sight the top of a tree. From the top of the building, 150 metres above ground level, I have to look down at an angle of 50o below the horizontal to sight the tree top. a How high is the tree? b How far from the building is this tree? 8 cm

8

Q

Find the measure of angle PQR in the rectangular box shown.

4 cm

P

7 cm R

9 Two observation posts are 12 km apart at A and B. A third observation post C is located such that angle CAB is 42o and angle CBA is 67o . Find the distance of C from both A and B.

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10 Stan and Olga are considering buying a sheep farm. A surveyor has supplied them with the given accurate sketch. Find the area of the property, giving your answer in: a km2 b hectares.

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11 Thabo and Palesa start at point A. They each walk in a straight line at an angle of 120o to each other. Thabo walks at 6 km h¡1 and Palesa walks at 8 km h¡1 . How far apart are they after 45 minutes? A

12 The cross-section design of the kerbing for a driverless-bus roadway is shown opposite. The metal strip is inlaid into the concrete and is used to control the direction and speed of the bus. Find the width of the metal strip.

30° 38° 200 mm 110° B

C

D metal strip

13 An orienteer runs for 4 12 km, then turns through an angle of 32o and runs for another 6 km. How far is she from her starting point? 14 Sam and Markus are standing on level ground 100 metres apart. A large tree is due North of Markus and on a bearing of 065o from Sam. The top of the tree appears at an angle of elevation of 25o to Sam and 15o to Markus. Find the height of the tree. 15 A helicopter A, flying at an altitude of 4000 m, observes two ships B and C. B is 23:8 km from the helicopter and C is 31:9 km from it. The angle of view from the helicopter to B and C (angle BAC) is 83:6o . How far are the ships apart?

REVIEW SET 11A 1 Determine the value of x: a 11 cm

b 15 km 72°

13 cm

17 km 19 cm

x°

x km

2 Find the value of x: a 13 cm

x cm

b 14 cm

11 cm

47°

x°

21 cm

19 cm

3 Find the unknown side and angles:

4 Find the area of quadrilateral ABCD:

C

A 110° 7 cm

A 9.8 cm 11 cm

B

40°

D

74°

11 cm

16 cm

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5 A vertical tree is growing on the side of a hill with slope of 10o to the horizontal. From a point 50 m downhill from the tree, the angle of elevation to the top of the tree is 18o . Find the height of the tree. 6 From point A, the angle of elevation to the top of a tall building is 20o . On walking 80 m towards the building the angle of elevation is now 23o . How tall is the building? 7 Peter, Sue and Alix are sea-kayaking. Peter is 430 m from Sue on a bearing of 113o while Alix is on a bearing of 203o and a distance 310 m from Sue. Find the distance and bearing of Peter from Alix. 8 A family in Germany drives at 140 km h¡1 for 45 minutes on a bearing of 032o and then 180 km h¡1 for 40 minutes on a bearing 317o . Find the distance and bearing of the car from its starting point. 9 You are given details of a triangle such that you could use either the cosine rule or the sine rule to find an unknown. Which rule should you use? Explain your answer. 10 Kady was asked to draw the illustrated triangle exactly. a Use the cosine rule to find x. b What should Kady’s response be?

7 cm

8 cm 60° x cm

11 Soil contractor Frank was given the following dimensions over the telephone: The triangular garden plot ABC has angle CAB measuring 44o , [AC] is 8 m long and [BC] is 6 m long. Soil to a depth of 10 cm is required. a Explain why Frank needs extra information from his client. b What is the maximum volume of soil needed if his client is unable to supply the necessary information?

REVIEW SET 11B 1 Find the measure of angle EDG:

F

G H

E

3m

B

C 4m

A 3 km

A

2

D

40° P

B

7 km

D

Hikers Andrew and Brett take separate trails from their starting point P to get to their destination at D. They walk at an angle of 40o apart from each other as shown, and camp overnight at positions A and B respectively. How far does Brett have to walk the next day to reach the destination?

100°

8 km

6m

3 Consider the kite ABCD alongside: b = ABC. b a Use the cosine rule to show that ADC b = BAC. b b Use the sine rule to show that DAC

D A

C

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4 A boat sailing from A to B travels in a straight line until the captain realises he is off course. The boat is turned through an angle of 60o, then travels another 10 km to B. The trip would have been 4 km shorter if the boat had gone straight from A to B. How far did the boat travel?

60° B

C

10 km A

5 At 1 pm, runner A runs at 14 km h¡1 on a bearing of 25o , while runner B starting at the same point as A runs at 12 km h¡1 on a bearing of 97o . a At what time will A and B be 20 km apart? b What will be the bearing of B from A at this time? 6 Triangle ABC has [AB] of length 5 m, [AC] of length d m, C [BC] of length x m, and angle ABC measures 20o . a Find an expression for d2 in terms of x. dm b By using the fact that, for d > 0, d is minimised when d2 is minimised, find the exact value of x which A minimises d. b is a right angle. c Hence, show that d is minimised when BCA

B

20° 5m

12 m

a Explain why cos(180o ¡ µ) = ¡ cos µ. b For the given quadrilateral: i Show that 300 cos do ¡ 192 cos bo = 117. ii If b + d = 180, find the values of b and d. iii Hence, find the values of a and c.

7

xm

a°

b°

10 m

8m

d°

c° 15 m

a For the quadratic function y = ¡x2 +12x¡20, find the maximum or minimum value and the corresponding value of x: B b In triangle ABC, AB = y, BC = x, AC = 8, and the perimeter of the triangle is 20. y x i Write y in terms of x: 2 ii Use the cosine rule to write y in terms of x and q A cos µ: C 8 3x ¡ 10 . iii Hence show that cos µ = 2x c If the area of the triangle is A, show that A2 = 20(¡x2 + 12x ¡ 20): d Hence, find the maximum area of the triangle and the triangle’s shape when this occurs.

8

Q 9 The given figure shows quadrilateral PQRS 3 7 which has been divided into two triangles by the P f diagonal [QS]. PQ = 3, QR = 7, PS = 6, 32° b = 32o . Qb PS = Á and QRS 6 a Find QS in terms of cos Á. S b If Á = 50o : b i find the possible values of RSQ ii given that Rb SQ is acute, find the perimeter of the quadrilateral.

R

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c Find the area of quadrilateral PQRS if Á = 50o .

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Chapter

Advanced trigonometry Contents:

A B C D E F G H I J K L M

Observing periodic behaviour The sine function Modelling using sine functions The cosine function The tangent function Trigonometric equations Using trigonometric models Reciprocal trigonometric functions Trigonometric relationships Compound angle formulae Double angle formulae Trigonometric equations in quadratic form Trigonometric series and products

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Review Set 12A Review Set 12B Review Set 12C Review Set 12D

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ADVANCED TRIGONOMETRY (Chapter 12)

INTRODUCTION Periodic phenomena occur all the time in the physical world. For example, in: ² ² ² ² ² ²

seasonal variations in our climate variations in average maximum and minimum monthly temperatures the number of daylight hours at a particular location variations in the depth of water in a harbour due to tidal movement the phases of the moon animal populations.

These phenomena illustrate variable behaviour which is repeated over time. The repetition may be called periodic, oscillatory or cyclic in different situations. In this topic we will consider how trigonometric functions can be used to model periodic phenomena. We will then extend our knowledge of the trigonometric functions by considering formulae that connect them.

OPENING PROBLEM A Ferris wheel rotates at a constant speed. The wheel’s radius is 10 m and the bottom of the wheel is 2 m above ground level. From a point in front of the wheel, Andrew is watching a green light on the perimeter of the wheel. Andrew notices that the green light moves in a circle. He estimates how high the light is above ground level at two second intervals and draws a scatterplot of his results. ² What does his scatterplot look like? ² Could a known function be used to model the data? ² How could this function be used to find the light’s position at any point in time? ² How could this function be used to find the times when the light is at its maximum and minimum heights? ² What part of the function would indicate the time interval over which one complete cycle occurs? Click on the icon to visit a simulation of the Ferris wheel. You will be able to view the light on the Ferris wheel: ² from a position in front of the wheel ² from a side-on position ² from above the wheel.

DEMO

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Observe the graph of height above (or below) the wheel’s axis as the wheel rotates at a constant rate.

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OBSERVING PERIODIC BEHAVIOUR

Consider the table below which shows the mean monthly maximum temperature (o C) for Cape Town, South Africa. Month

Jan

Temp

28

Feb

Mar

27

25 12

Apr

May

22

18 12

Jun 16

Jul

Aug

15

16

Sep

Oct

Nov

Dec

18

21 12

24

26

T, Temp (°C)

The data is graphed alongside using a scatterplot, assigning January = 1, February = 2 etc., for the 12 months of the year.

30 20

Note: The points are not joined as interpolation has no meaning here.

(10,¡21\Qw_\)

10

t (months) 3

6

9

12 JAN

JAN

The temperature shows a variation from an average of 28o C in January through a range of values across the months and the cycle will repeat itself for the next 12 months. It is worthwhile noting that later we will be able to establish a function which approximately fits this set of points. T, Temp (°C) 30 20 10

t (months) 3

6

9

12 15 JAN

JAN

18

21

24

HISTORICAL NOTE

direction of rotation

+

voltage

lines of force

90°

-

180°

270°

360°

In 1831 Michael Faraday discovered that an electric current was generated by rotating a coil of wire in a magnetic field. The electric current produced showed a voltage which varied between positive and negative values as the coil rotated through 360 o.

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Graphs with this basic shape where the cycle is repeated over and over are called sine waves.

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GATHERING PERIODIC DATA Data on a number of periodic phenomena can be found online or in other publications. For example: ²

Maximum and minimum monthly temperatures can be found at http://www.bom.gov.au/silo/

²

Tidal details can be obtained from daily newspapers or http://tidesandcurrents.noaa.gov or http://www.bom.gov.au/oceanography

TERMINOLOGY USED TO DESCRIBE PERIODICITY A periodic function is one which repeats itself over and over in a horizontal direction. The period of a periodic function is the length of one repetition or cycle. f(x) is a periodic function with period p , f (x + p) = f (x) for all x, and p is the smallest positive value for this to be true. Use a graphing package to examine the function: f : x 7! x ¡ [x] where [x] is “the largest integer less than or equal to x”.

GRAPHING PACKAGE

Is f(x) periodic? What is its period? A cycloid is another example of a periodic function. It is the curve traced out by a point on a circle as the circle moves along a flat surface. However, the cycloid function cannot be written as y = :::::: or f (x) = :::::: DEMO

horizontal flat surface

In this course we are mainly concerned with periodic phenomena which show a wave pattern when graphed. the wave

principal axis

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The wave oscillates about a horizontal line called the principal axis or mean line. A maximum point occurs at the top of a crest and a minimum point at the bottom of a trough. The amplitude is the distance between a maximum (or minimum) point and the principal axis.

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maximum point amplitude

principal axis

period

minimum point

EXERCISE 12A 1 For each set of data below, draw a scatterplot and decide whether or not the data exhibits approximately periodic behaviour. a

x y

b

x y

c

x y

d

x y

0 0 0 4

1 1 1 1

2 0

0 0 0 0

2 1:4 3 1

0:5 1:9 2 4:7

3 1

4 0

5 ¡1

1:5 4:5

2:0 4:7

6 ¡1:4

7 ¡1

3:0 3:4

3:5 2:4

8 0

9 1

10 1:4

11 1

12 0

4 4

1:0 3:5 3 3:4

4 1:7

5 2:1

2:5 4:3 6 5:2

7 8:9

8 10:9

9 10:2

10 8:4

12 10:4

2 The following tabled values show the height above the ground of a point on a bicycle wheel as the bicycle is wheeled along a flat surface. Distance travelled (cm) Height above ground (cm)

0 0

20 6

40 23

60 42

80 57

100 120 140 160 180 200 64 59 43 23 7 1

Distance travelled (cm) 220 240 260 280 300 320 340 360 380 400 Height above ground (cm) 5 27 40 55 63 60 44 24 9 3 a Plot the graph of height against distance. b Is the data periodic? If so, estimate: i the equation of the principal axis ii the maximum value iii the period iv the amplitude. c Is it reasonable to fit a curve to this data, or should we leave it as discrete points? 3 Which of these graphs show periodic behaviour? y a b 1

2

3

4

6 x

5

3

y

c

y

6

9

12

15

18 x

y

d 2

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THE SINE FUNCTION

In previous studies of trigonometry we have only considered right angled triangles, or static situations where an angle µ is fixed. However, when an object moves in a circle, the situation is dynamic with the angle between the radius OP and the horizontal axis continually changing. y

Consider again the Opening Problem in which a Ferris wheel of radius 10 m revolves at constant speed. The height of P, the point representing the green light on the wheel relative to the principal axis O, can be determined using right angled triangle trigonometry. As sin µ =

P 10

h

q

x 10

h , then h = 10 sin µ: 10

As time goes by µ changes and so does h. So, h is a function of µ, but more importantly h is a function of time t. Suppose the Ferris wheel observed by Andrew takes 100 seconds for a full revolution. The graph below shows the height of the light above or below the principal axis against the time in seconds. 10

height (metres)

DEMO

50

100

time (seconds)

-10

We observe that the amplitude is 10 metres and the period is 100 seconds. The family of sine curves can have different amplitudes and different periods. We will examine such families in this section.

THE BASIC SINE CURVE If we project the values of sin µ from the unit circle to the set of axes on the right we obtain the graph of y = sin x. y

y 1 y = sin x

90°

180° p

p 2

270° 3p 2

360° 2p

x

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The wave of course can be continued beyond 0 6 x 6 2¼. y 1 90°

180° p

p 2

270°

360° 2p

3p 2

540° x 3p y = sin x

-1

We expect the period to be 2¼, since the Ferris wheel repeats its positioning after one full revolution. The maximum value is 1 and the minimum is ¡1 as ¡1 6 y 6 1 on the unit circle. The amplitude is 1. GRAPHING TI

PACKAGE

Use your graphics calculator or graphing package to obtain the graph of y = sin x to check these features.

C

When patterns of variation can be identified and quantified in terms of a formula or equation, predictions may be made about behaviour in the future. Examples of this include tidal movement which can be predicted many months ahead, and the date of a future full moon.

THE FAMILY y = Asin¡x


GRAPHING

1 Use technology to graph on the same set of axes: PACKAGE a y = sin x and y = 2 sin x b y = sin x and y = 0:5 sin x c y = sin x and y = ¡ sin x (A = ¡1) If using a graphics calculator, make sure that the mode is set in radians and that your viewing window is appropriate. 2 For each of y = sin x, y = 2 sin x, y = 0:5 sin x, y = ¡ sin x record the maximum and minimum values and state the period and amplitude. 3 How does A affect the function y = A sin x? 4 State the amplitude of: a y = 3 sin x

b

y=

p 7 sin x

c

y = ¡2 sin x

THE FAMILY y = sin¡Bx, B > 0


1 Use technology to graph on the same set of axes: a y = sin x and y = sin 2x b y = sin x and y = sin( 12 x) 2 For each of y = sin x, y = sin 2x, y = sin( 12 x) record the maximum and minimum values and state the period and amplitude. 3 How does B affect the function y = sin Bx?

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y = sin( 13 x)

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4 State the period of: a y = sin 3x

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GRAPHING PACKAGE

d

y = sin Bx

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From the previous investigations you should have observed that: ²

in y = A sin x,

j A j determines the amplitude

²

in y = sin Bx, B > 0,

B affects the period and the period is

Recall j x j is the modulus of x, or size of x ignoring its sign.

2¼ : B

The modulus sign ensures that the final answer is non-negative. This must be so for amplitudes.

Example 1 Without using technology sketch the graphs of: a y = 2 sin x b y = ¡2 sin x for 0 6 x 6 2¼. a

The amplitude is 2 and the period is 2¼: y 2 3p 2

p 2

2p

x

p

-2

We place the 5 points as shown and fit the sine wave to them. b

The amplitude is 2, the period is 2¼, and it is the reflection of y = 2 sin x in the x-axis. y 2

y¡=¡-2sin¡x 3p 2

p 2

2p

x

p

y¡=¡2sin¡x

-2

Example 2 Without using technology sketch the graph of y = sin 2x, 0 6 x 6 2¼. 2¼ 2

The period is

= ¼:

As sin 2x has half the period of sin x, the first maximum is at ¼2 not ¼4 .

So, for example, the maximum values are ¼ units apart. y 3p 2

2p

p

p 2

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EXERCISE 12B.1 1 Without using technology sketch the graphs of the following for 0 6 x 6 2¼: a

y = ¡3 sin x

b

y = 3 sin x

c

y=

3 2

sin x

d

y = ¡ 32 sin x

2 Without using technology sketch the graphs of the following for 0 6 x 6 3¼: ¡ ¢ c y = sin(¡2x) a y = sin 3x b y = sin x2 3 State the period of: a y = sin 4x

b

c

y = sin(¡4x)

y = sin

¡x¢ 3

d

4 Find B given that the function y = sin Bx, B > 0 has period: c 12¼ d 4 e a 5¼ b 2¼ 3

y = sin(0:6x)

100

5 Use a graphics calculator or graphing package to help you graph, for 0o 6 x 6 720o : 1 a y = 2 sin x + sin 2x b y = sin x + sin 2x + sin 3x c y= sin x 6 Use a graphing package or graphics calculator to graph: a f (x) = sin x +

sin 3x sin 5x + 3 5

b f (x) = sin x +

sin 3x sin 5x sin 7x sin 9x sin 11x + + + + 3 5 7 9 11

Predict the graph of f (x) = sin x +

GRAPHING PACKAGE

sin 3x sin 5x sin 7x sin 1001x + + + :::::: + 3 5 7 1001

INVESTIGATION 3 THE FAMILIES y = sin(x ¡ C) AND y = sin¡x + D What to do: 1 Use a b c

technology to graph on the same set of axes: y = sin x and y = sin(x ¡ 2) y = sin x and y = sin(x + 2) y = sin x and y = sin(x ¡ ¼3 )

GRAPHING PACKAGE

2 For each of y = sin x, y = sin(x ¡ 2), y = sin(x + 2), y = sin(x ¡ ¼3 ) record the maximum and minimum values and state the period and amplitude. 3 What transformation moves y = sin x to y = sin(x ¡ C)? 4 Use technology to graph on the same set of axes: a y = sin x and y = sin x + 3 b y = sin x and y = sin x ¡ 2 5 For each of y = sin x, y = sin x + 3 and y = sin x ¡ 2 record the maximum and minimum values and state the period and amplitude. 6 What transformation moves y = sin x to y = sin x + D?

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From Investigation 3 we observe that: ² ²

y = sin(x ¡ C) is a horizontal translation of y = sin x through C units. y = sin x + D is a vertical translation of y = sin x through D units. h i C . y = sin(x ¡ C) + D is a translation of y = sin x through vector D

²

Example 3 On the same set of axes graph for 0 6 x 6 4¼: a y = sin x and y = sin(x ¡ 1) b y = sin x and y = sin x ¡ 1 a

y

1

1

1

p

2p

1

4p

3p

b

1

y¡=¡sin¡x

1

-1

x y¡=¡sin(¡ x¡-¡1)

y

-1

-1

p

2p

-1

4p

3p -1

-1

-2

x y¡=¡sin¡x y¡=¡sin¡x¡-¡1

THE GENERAL SINE FUNCTION y = A sin B(x ¡ C) + D affects amplitude

affects period

is called the general sine function.

affects horizontal translation

affects vertical translation

The principal axis of the general sine function is y = D. The period of the general sine function is

2¼ . B

Consider y = 2 sin 3(x ¡ ¼4 ) + 1: It is a translation of y = 2 sin 3x under

·¼¸ 4

1

.

So, starting with y = sin x we would:

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y = 2 sin x, then y = 2 sin 3x, then ¢ ¡ y = 2 sin 3 x ¡ ¼4 + 1:

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double the amplitude to produce divide the period by 3 to produce ·¼¸ 4 to produce translate by

5

² ²

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EXERCISE 12B.2 1 Draw sketch graphs of: a y = sin x ¡ 2 d

y = sin(x ¡ 2)

b e

y = sin x + 2

¼ 4)

y = sin(x +

c

y = sin(x + 2)

f

y = sin(x ¡ ¼6 ) + 1

2 Check your answers to 1 using technology.

3 State the period of: a y = sin 5t

b

y = sin

GRAPHING PACKAGE

¡t¢

c

4

y = sin(¡2t)

4 Find B in y = sin Bx if B > 0 and the period is: ¼ c 100¼ a 3¼ b 10 5 State the transformation(s) which maps: a y = sin x onto y = sin x ¡ 1 c

y = sin x onto y = 2 sin x

d

50

b

y = sin x onto y = sin(x ¡ ¼4 )

d

e

y = sin x onto y =

sin x

f

y = sin x onto y = sin 4x ¡ ¢ y = sin x onto y = sin x4

g

y = sin x onto y = ¡ sin x

h

y = sin x onto y = ¡3 + sin(x + 2)

i

y = sin x onto y = 2 sin 3x

j

y = sin x onto y = sin(x ¡ ¼3 ) + 2

C

1 2

MODELLING USING SINE FUNCTIONS

Sine functions can be useful for modelling certain biological and physical phenomena in nature which are approximately periodic.

MEAN MONTHLY TEMPERATURE Consider again the mean monthly maximum temperature (o C) for Cape Town: Month

Jan

Temp

28

Feb

Mar

27

25 12

Apr

May

22

18 12

Jun

Jul

16

15

Aug 16

Sep

Oct

Nov

Dec

18

21 12

24

26

The graph over a two year period is shown below: 40 30

T, temperature (°C) A

20 10

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We attempt to model this data using the general sine function y = A sin B(x ¡ C) + D, or in this case T = A sin B(t ¡ C) + D. 2¼ = 12 and ) B = ¼6 : The period is 12 months, so B The amplitude =

max ¡ min 28 ¡ 15 ¼ ¼ 6:5, so A ¼ 6:5 : 2 2

The principal axis is midway between max. and min., so D ¼

28 + 15 ¼ 21:5 . 2

So, the model is T ¼ 6:5 sin ¼6 (t ¡ C) + 21:5 for some constant C. We notice that point A on the original graph lies on the principle axis and is a point at which we are starting a new period. ) since A is at (10, 21:5), C = 10. The model is therefore T ¼ 6:5 sin ¼6 (t ¡ 10) + 21:5 and we can superimpose it on the original data as follows. 40

T, temperature (°C)

30 20 10

t (months)

0

Jan

Mar

May

Nov

Sep

Jul

Jan

Mar

May

Nov

Sep

Jul

TIDAL MODELS The tides at Juneau, Alaska were recorded over a two day period. The results are shown in the table opposite:

Day 1 Day 2

high tide low tide high tide low tide

1.18 pm 6.46 am, 7.13 pm 1.31 am, 2.09 pm 7.30 am, 7.57 pm

Suppose high tide corresponds to height 1 and low tide to height ¡1. Plotting these times with t being the time after midnight before the first low tide, we get:

1

12 hrs 13 min

Tide height, H

12 hrs 38 min

X t 6 am

12 noon

6 pm

12 pm

6 am

12 noon

6 pm

midnight -1 12 hrs 27 min

12 hrs 17 min

12 hrs 27 min

We attempt to model this periodic data using H = A sin B(t ¡ C) + D.

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The principal axis is H = 0, so D = 0. The amplitude is 1, so A = 1.

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The graph shows that the ‘average’ period is about 12 hours 24 min ¼ 12:4 hours. 2¼ 2¼ 2¼ But the period is . ) ¼ 12:4 and so B ¼ ¼ 0:507 . B B 12:4 The model is now H ¼ sin 0:507(t ¡ C) for some constant C. We find point X which is midway between the first minimum and the following maximum, 6:77 + 13:3 ¼ 10:0 . ) C¼ 2 So, the model is H ¼ sin 0:507(t ¡ 10:04): Below is our original graph of seven plotted points and our model which attempts to fit them. H¡¼¡sin\\0"507(t-10"04)

H 1

6 pm 6 am

12 noon

6 pm

12 pm

6 am

t

12 noon

-1

Use your graphics calculator to check this result. Times must be given in hours after midnight, i.e., (6:77, ¡1), (13:3, 1), (19:22, ¡1), etc.

MODELLING

TI C

EXERCISE 12C 1 Below is a table which shows the mean monthly maximum temperatures (o C) for a city in Greece. Month Jan Feb Mar Apr May Jun July Aug Sept Oct Nov Dec o Temperature ( C) 15 14 15 18 21 25 27 26 24 20 18 16 a Use a sine function of the form T ¼ A sin B(t ¡ C) + D to model the data. Find good estimates of the constants A, B, C and D without using technology. Use Jan ´ 1, Feb ´ 2, etc. b Use technology to check your answer to a. How well does your model fit? 2 The data in the table shows the mean monthly temperatures for Christchurch. Month

Jan Feb Mar Apr May Jun July Aug Sept Oct Nov Dec o

Temperature ( C) 15

16

14 12

12

10

7 12

7

7 12

8 12

10 12 12 12

14

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a Find a sine model for this data in the form T ¼ A sin B(t ¡ C) + D. Assume Jan ´ 1, Feb ´ 2, etc. Do not use technology. b Use technology to check your answer to a.

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3 At the Mawson base in Antarctica, the mean monthly temperatures for the last 30 years are as follows: Month Jan Feb Mar Apr May Jun July Aug Sept Oct Nov Dec Temperature (o C) 0 ¡4 ¡10 ¡15 ¡16 ¡17 ¡18 ¡19 ¡17 ¡13 ¡6 ¡1 Find a sine model for this data using your calculator. Use Jan ´ 1, Feb ´ 2, etc. How appropriate is the model? 4 Some of the largest tides in the world are observed in Canada’s Bay of Fundy. The difference between high and low tides is 14 metres and the average time difference between high tides is about 12:4 hours. a Find a sine model for the height of the tide H in terms of the time t. b Sketch the graph of the model over one period. 5 Revisit the Opening Problem on page 284. The wheel takes 100 seconds to complete one revolution. Find the sine model which gives the height of the light above the ground at any point in time. Assume that at time t = 0, the light is at its lowest point.

D

10 m

2m

THE COSINE FUNCTION We return to the Ferris wheel to see the cosine function being generated.

DEMO

y

Click on the icon to inspect a simulation of the view from above the wheel.

10 q

The graph being generated over time is a cosine function. d This is no surprise as cos µ = 10 and so d = 10 cos µ.

x d

y

Now view the relationship between the sine and cosine functions. Notice that the functions are identical in shape, but the cosine function is ¼2 units left of the sine function under a horizontal translation.

y¡=¡sin¡x -p 2

p

p 2

¡ cos x = sin x +

This suggests that

-p 2

¼ 2

3p 2 -p 2

¢ .

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Use your graphing package or graphics calculator to check this by graphing ¢ ¡ y = cos x and y = sin x + ¼2 .

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2p

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y¡=¡cos¡x

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Example 4 ¢ ¡ On the same set of axes graph y = cos x and y = cos x ¡ ¼3 . ¢ ¡ y = cos x ¡ ¼3 comes from y = cos x under a horizontal translation through y

y = cos x

1 -p

p 3

-2p

p 3

Qw_

p

x

2p

p 3

-1

p 3

¼ 3.

y = cos&x - p3* y

EXERCISE 12D 1 Given the graph of y = cos x, sketch the graphs of:

1 x

-p

p

2p

-1

a d g j

y y y y

b e h k

= cos x + 2 = cos(x + ¼6 ) = ¡ cos x = cos 2x

y = cos x ¡ 1 y = 23 cos x y = cos(x ¡ ¼6 ) + 1 ¡ ¢ y = cos x2

2 Without graphing them, state the periods of: ¡ ¢ a y = cos 3x b y = cos x3

= cos(x ¡ ¼4 ) = 32 cos x = cos(x + ¼4 ) ¡ 1 = 3 cos 2x

c f i l

y y y y

c

y = cos

¡¼ ¢ 50 x

3 The general cosine function is y = A cos B(x ¡ C) + D. State the geometrical significance of A, B, C and D. 4 For the following graphs, find the cosine function representing them: a b c y y y 3

2

x

2p

5

6

1

-2

2p

E

4p

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THE TANGENT FUNCTION

Consider the unit circle diagram given. P(cos µ, sin µ) is a point which is free to move around the circle. In the first quadrant we extend [OP] to meet the tangent at A(1, 0) so the intersection occurs at Q. As P moves, so does Q. The position of Q relative to A is defined as the tangent function. Notice that ¢s ONP and OAQ are equiangular and therefore similar.

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N A(1, 0)

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AQ NP = OA ON

Consequently

AQ sin µ = 1 cos µ

i.e.,

which suggests that tan µ =

sin µ . cos µ

y

P

The question arises: “If P does not lie in the first quadrant, how is tan µ defined?”

q A(1, 0)

x

For µ obtuse, since sin µ is positive and cos µ is sin µ negative, tan µ = is negative. cos µ As before, [PO] is extended to meet the tangent at A at Q(1, tan µ).

Q(1, tan¡q)

y q

Q(1, tan¡q) A(1, 0)

For µ in quadrant 3, sin µ and cos µ are both negative and so tan µ is positive. This is clearly demonstrated as Q is back above the x-axis.

x

P

y

For µ in quadrant 4, sin µ is negative and cos µ is positive. So, tan µ is negative.

q A(1, 0)

x

P Q(1, tan¡q)

DISCUSSION ²

What is tan µ when P is at (0, 1)?

²

What is tan µ when P is at (0, ¡1)?

EXERCISE 12E.1 1 Use your calculator to find the value of: b tan 15o a tan 0o e tan 35o f tan 45o

tan 20o tan 50o

c g

d h

tan 25o tan 55o

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2 Explain why tan 45o = 1 exactly.

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THE GRAPH OF y = tan¡x

y

3

The graph of y = tan x is

x

DEMO

-p - p2

p 2

3p 2

p

2p

5p 2

-3

DISCUSSION ² ²

Is the tangent function periodic? If so, what is its period? For what values of x does the graph not exist?¡ What physical characteristics are shown near these values?¡ Explain why these values must occur when cos x = 0:

² ² ²

Discuss how to find the x-intercepts of y = tan x. What must tan(x ¡ ¼) simplify to? How many solutions does the equation tan x = 2 have?

EXERCISE 12E.2 a Use a transformation approach to sketch the graphs of these functions, x 2 [0, 3¼]: ii y = ¡ tan x iii y = tan 2x i y = tan(x ¡ ¼2 )

1

b Use technology to check your answers to a. Look in particular for: ² asymptotes ²

x-axis intercepts.

GRAPHING PACKAGE

2 Use the graphing package to graph, on the same set of axes: a y = tan x and y = tan(x ¡ 1) b y = tan x and y = ¡ tan x ¡x¢ c y = tan x and y = tan 2

GRAPHING PACKAGE

Describe the transformation which moves the first curve to the second in each case.

3 What is the period of: a y = tan x

F

b

y = tan 2x

c

y = tan nx?

TRIGONOMETRIC EQUATIONS

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Linear equations such as 2x + 3 = 11 have exactly one solution. Quadratic equations of the form ax2 + bx + c = 0, a 6= 0 have at most two real solutions. Trigonometric equations generally have infinitely many solutions unless a restrictive domain such as 0 6 x 6 3¼ is given. We will examine solving trigonometric equations using: ² preprepared graphs ² technology ² algebraic methods.

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¼ For the Ferris wheel Opening Problem the model is H = 10 sin 50 (t ¡ 25) + 12.

t = 50

We can easily check this by substituting t = 0, 25, 50, 75 ¡ ¢ H(0) = 10 sin ¡ ¼2 + 12 = ¡10 + 12 = 2 X

10 m

t = 75

H(25) = 10 sin 0 + 12 = 12 X ¡ ¢ H(50) = 10 sin ¼2 + 12 = 22 X etc.

t = 25 10 m

2m

However, we may be interested in the times when the light is some other height above the ground, for example 16 m. We would then need to solve the equation ¼ (t ¡ 25) + 12 = 16 which is called a sine equation. 10 sin 50

GRAPHICAL SOLUTION OF TRIGONOMETRIC EQUATIONS Sometimes simple sine or cosine graphs are available on grid paper. In such cases we can estimate solutions straight from the graph. For example, we could use a graph to find approximate solutions for trigonometric equations such as cos µ = 0:4 for 0 6 µ 6 10 radians. y 1

A

B

1

p 3

2

4

2p 6

5

p 2

-1

C 3p 7

8

3p 2

9

y¡¡=¡0.4 q

10

5p 2

y = cos¡q

y = 0:4 meets y = cos µ at A, B and C and hence µ ¼ 1:2, 5:1 or 7:4 . So, the solutions of cos µ = 0:4 for 0 6 µ 6 10 radians are 1:2, 5:1 and 7:4 .

EXERCISE 12F.1 1

y

y¡=¡sin¡x

1 0.5

x 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

-0.5 -1

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Use the graph of y = sin x to find, correct to 1 decimal place, the solutions of: a sin x = 0:3 for 0 6 x 6 15 b sin x = ¡0:4 for 5 6 x 6 15

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2 y y=cos¡x

1 0.5

x 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

-0.5 -1

Use the graph of y = cos x to find, to 1 decimal place, approximate solutions of: a cos x = 0:4, x 2 [ 0, 10 ] b cos x = ¡0:3, x 2 [ 4, 12 ] 3 y

y¡=¡sin2 ¡ x

1 0.5 x 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

-0.5 -1

Use the graph of y = sin 2x to find, correct to 1 decimal place, the solutions of: a sin 2x = 0:7, x 2 [ 0, 16 ] b sin 2x = ¡0:3, x 2 [ 0, 16 ]

4 The graph of y = tan x is illustrated.

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a Use the graph to find estimates of: i tan 1 ii tan 2:3 b Check your answers with a calculator. c Find, correct to 1 decimal place, the solutions of: i tan x = 2 for 06x68 ii tan x = ¡1:4 for 26x67

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SOLVING TRIGONOMETRIC EQUATIONS USING TECHNOLOGY Trigonometric equations may be solved using either a graphing package or a graphics display calculator (gdc). GRAPHING PACKAGE

When using a graphics calculator make sure that the mode is set to radians.

TI C

Example 5 Solve 2 sin x ¡ cos x = 4 ¡ x for 0 6 x 6 2¼. We need to use window settings just larger than the domain. Xmin = ¡ ¼6

In this case,

Xmax =

13¼ 6

Xscale =

¼ 6

The grid facility on the gdc can also be helpful, particularly when a sketch is required. Using the appropriate function on the gdc gives the following solutions: x = 1:82, 3:28, 5:81 (3 s.f.)

EXERCISE 12F.2 1 Solve each of the following for 0 6 x 6 2¼: a c

sin(x + 2) = 0:0652 µ 2¶ x x tan = x2 ¡ 6x + 1 10

b

sin2 x + sin x ¡ 1 = 0

d

2 sin(2x) cos x = ln x

cos(x ¡ 1) + sin(x + 1) = 6x + 5x2 ¡ x3

2 Solve for x, ¡2 6 x 6 6:

SOLVING TRIGONOMETRIC EQUATIONS ALGEBRAICALLY Using a calculator we get approximate decimal or numerical solutions to trigonometric equations. Sometimes exact solutions are needed in terms of ¼, and these arise when the solutions are multiples of ¼6 or ¼4 . Exact solutions obtained using algebra are called analytical solutions. y

³ ´ ¡ p12 ; p12

(0, 1)

³

p1 ; p1 2 2

³ p ´ ¡ 12 ; 23

´

x (1, 0)

(-1, 0)

³ ´ ¡ p12 ; ¡ p12

³

p1 ; ¡ p1 2 2

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(0, 1)

³

p ´ 1 3 2; 2

³p

3 1 2 ;2

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´

x (1, 0) ´ ³p 3 1 2 ; ¡2

(-1, 0) ³ p ´ ¡ 23 ; ¡ 12

(0,-1)

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³ p ´ ¡ 23 ; 12

100

Reminder:

³ (0,-1)

p ´ 1 3 2;¡ 2

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Example 6 Use the unit circle to find the exact solutions of x, 0 6 x 6 3¼ for: ¢ ¡ b sin 2x = ¡ 12 c sin x ¡ ¼6 = ¡ 12 a sin x = ¡ 12 sin x = ¡ 12 , so from the unit circle 7¼ ¾ 6 x = 11¼ + k2¼, k an integer

a

y

x

6

)

7¼ 6 ,

11¼ 6 ,

19¼ 6

k=0

k=0

k=1

x=

is too big

7p 6

11p 6

@=-\Qw_

Substituting k = 1, 2, 3, ..... gives answers outside the required domain. Likewise k = ¡1, ¡2, ..... gives answers outside the required domain. 11¼ ) there are two solutions: x = 7¼ 6 or 6 . sin 2x = ¡ 12

b

is solved in exactly the same way: ¾ 7¼ 6 2x = 11¼ + k2¼, k an integer 6

)

x=

)

x=

7¼ 11¼ 19¼ 23¼ 31¼ 35¼ 12 , 12 , 12 , 12 , 12 , 12

k=0

k=1

or 2¼

which is three solutions.

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0

Don’t forget to try k¡=¡¡1, ¡2, etc. as sometimes we get solutions from them.

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k = ¡1

100

50

is too big,

75

4¼ 3

25

k=0

0

10¼ 3

5

0

to both sidesg

2¼,

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x=

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fadding

+ k2¼

2¼

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x=

6

¾

8¼ 6

75

)

¼ 6

So, x = 0,

0

fobtained by letting k = 0, 1, 2g

is also solved in the same way: ¾ 7¼ 6 = 11¼ + k2¼

25

x¡

5

fdivide each term by 2g

+ k¼

11¼ 12

sin(x ¡ ¼6 ) = ¡ 12

c

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7¼ 12

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Example 7 p ¡ 2 cos x ¡

Find exact solutions of

p ¡ 2 cos x ¡

Rearranging We recognise

p1 2

)

3¼ 4

x¡ )

If k = ¡1, If k = 1,

3¼ 4

¢

3¼ 4

¢

+ 1 = 0 for x 2 [ 0, 6¼ ].

¡ + 1 = 0, we find cos x ¡

4

x=

y

x

+ k2¼ -

x = ¡ ¼2 or 0: x=

= ¡ p12

¾

2¼

7¼ 2

¢

¼ 4)

as a special fraction (for multiples of 3¼ ¾ 4 = 5¼ + k2¼, k an integer 3¼ 2

3¼ 4

or 4¼:

If k = 0,

x=

If k = 2,

x=

1 2

» -0.7

3¼ 2 11¼ 2

or 2¼: or 6¼:

If k = 3, the answers are greater than 6¼. 3¼ 2 ,

So, the solutions are: x = 0,

2¼,

7¼ 2 ,

11¼ 2

4¼,

or 6¼:

Since the tangent function is periodic with period ¼ we see that tan(x + ¼) = tan x for all values of x. This means that equal tan values are ¼ units apart.

Example 8 ¢ ¡ Find exact solutions of tan 2x ¡ ¼3 = 1 for x 2 [¡¼, ¼ ] 2x ¡ )

¼ 3

x=

x=

¡ 17¼ 24 ,

¡ 5¼ 24 ,

7¼ 24 ,

19¼ 24

k = ¡2

k = ¡1

k=0

k=1

2x =

)

)

fsince tan ¼4 = 1g

¼ 4 + k¼ 7¼ 12 + k¼ 7¼ ¼ 24 + k 2

=

y

(

1 2

,

1 2

)

x p 4

Example 9 p 3 sin x = cos x for 0 6 x 6 2¼: p 3 sin x = cos x p sin x 1 ) fdividing both sides by 3 cos xg = p cos x 3

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- 3 2

5

(

100

x p 6

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1 2

75

3 2

25

( ,)

0

y

5

Find the exact solutions of

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EXERCISE 12F.3 1 List the possible solutions for x if k is an integer and: b x = ¡ ¼3 + k2¼, ¡2¼ 6 x 6 2¼ a x = ¼6 + k2¼, 0 6 x 6 6¼ ¡¼¢ c x = ¡ ¼2 + k¼, ¡4¼ 6 x 6 4¼ d x = 5¼ 6 + k 2 , 0 6 x 6 4¼ 2 Find the exact solutions of: a cos x = ¡ 12 , x 2 [ 0, 5¼ ] p c 2 cos x + 3 = 0, 0 6 x 6 3¼ ¡ ¢ e 2 sin x + ¼3 = 1, x 2 [¡3¼, 3¼ ]

2 sin x ¡ 1 = 0, ¡2¼ 6 x 6 2¼ ¢ 1 ¡ = 2 , x 2 [¡2¼, 2¼ ] cos x ¡ 2¼ 3 p ¢ ¡ 2 sin x ¡ ¼4 + 1 = 0, x 2 [ 0, 3¼ ]

b d f

p 3 Find the exact solutions of tan X = 3 in terms of ¼ only. Hence solve the equations: p ¢ p ¡ b tan 4x = 3 c tan2 x = 3 a tan x ¡ ¼6 = 3 4 Find the zeros of: a y = sin 2x between 0 and ¼ (inclusive) b y = sin(x ¡ ¼4 ) between 0 and 3¼ (inclusive) 5

a Use your graphics calculator to sketch the graphs of y = sin x and y = cos x on the same set of axes on the domain x 2 [ 0, 2¼ ]. b Find the x values of the points of intersection of the two graphs. c Confirm that these values are the solutions of sin x = cos x on x 2 [ 0, 2¼ ].

6 Find the exact solutions to these equations for 0 6 x 6 2¼ : a sin x = ¡ cos x b sin(3x) = cos(3x) c

sin(2x) =

p 3 cos(2x)

7 Check your answers to question 6 using a graphics calculator. Find the points of intersection of the appropriate graphs.

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USING TRIGONOMETRIC MODELS

Example 10

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The height h(t) metres of the tide above mean sea level on January 24th at Cape ¡ ¢ where t is the number Town is modelled approximately by h(t) = 3 sin ¼t 6 of hours after midnight. a Graph y = h(t) for 0 6 t 6 24: b When is high tide and what is the maximum height? c What is the height at 2 pm? d If a ship can cross the harbour provided the tide is at least 2 m above mean sea level, when is crossing possible on January 24?

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a

h(t) = 3 sin 3

h(t)

¡ ¼t ¢

has period =

6

2¼ ¼ 6

A

6

9

12

= 12 hours

and h(0) = 0

15

18

21

24

t

noon

-3

c

6 ¼

B

3

b

= 2¼ £

High tide is at 3 am and 3 pm. The maximum height is 3 m above the mean as seen at points A and B. ¢ ¡ ¼ 2:60 (3 sig figs) At 2 pm, t = 14 and h(14) = 3 sin 14¼ 6 So the tide is 2:6 m above the mean.

d 3 2

h(t)

A

3

t1

(14, 2.60) B

6

t2

9

12 t3

15

-3

We need to solve h(t) = 2

h¡=¡2

i.e., 3 sin

t4 18

21

24

t

¡ ¼t ¢

6 = 2: ¡ ¼X ¢ Using a graphics calculator with Y1 = 3 sin 6 and Y2 = 2

we obtain t1 = 1:39, t2 = 4:61, t3 = 13:39, t4 = 16:61 or you could trace across the graph to find these values. Now 1:39 hours = 1 hour 23 minutes, etc. ) can cross between 1:23 am and 4:37 am or 1:23 pm and 4:37 pm.

EXERCISE 12G 1 The population of grasshoppers after t weeks where 0 6 t 6 12 is estimated by ¡ ¢ P (t) = 7500 + 3000 sin ¼t 8 :

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What is: i the initial estimate ii the estimate after 5 weeks? What is the greatest population size over this interval and when does it occur? When is the population i 9000 ii 6000? During what time interval(s) does the population size exceed 10 000? ¡ ¢ 2 The model for the height of a light on a Ferris wheel is H(t) = 20 ¡ 19 sin 2¼t 3 , where H is the height in metres above the ground, and t is in minutes. a Where is the light at time t = 0? b At what time is the light at its lowest in the first revolution of the wheel? c How long does the wheel take to complete one revolution? d Sketch the graph of the function H(t) over one revolution.

5

a b c d

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3 The population of water buffalo is given by ¡ ¢ where t is the number of P (t) = 400 + 250 sin ¼t 2 years since the first estimate was made. a What was the initial estimate? b What was the population size after: i 6 months ii two years? c Find P (1). What is the significance of this value? d Find the smallest population size and when it first occurs. e Find the first time when the herd exceeded 500. 4 A paint spot X lies on the outer rim of the wheel of a paddle-steamer. The wheel has radius 3 m and as it rotates at a constant rate, X is seen entering the water every 4 seconds. H is the distance of X above the Hm bottom of the boat. At time t = 0, X is at its highest point. a Find a cosine model for H in the form H(t) = A cos B(t ¡ C) + D. b At what time does X first enter the water?

X

3m

water level 1m 1m bottom of the boat

5 Over a 28 day period, the cost per litre of petrol was modelled by C(t) = 9:2 sin ¼7 (t ¡ 4) + 107:8 cents L¡1 . a True or false? i “The cost per litre oscillates about 107:8 cents with maximum price $1:17:” ii “Every 14 days, the cycle repeats itself.” b What was the cost of petrol at day 7? c On what days was the petrol priced at $1:10 per litre? d What was the minimum cost per litre and when did it occur?

H

RECIPROCAL TRIGONOMETRIC FUNCTIONS

We define the reciprocal trigonometric functions cosec x, secant x and cotangent x as: csc x =

1 , sin x

sec x =

1 cos x

and

cot x =

1 cos x = tan x sin x

Note the important identities: tan2 x + 1 = sec2 x and 1 + cot2 x = csc2 x Start with sin2 x + cos2 x = 1

Proof:

sin2 x cos2 x 1 + = 2 2 cos x cos x cos2 x

)

fdividing each term by cos2 xg

) tan2 x + 1 = sec2 x

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To prove the second identity we instead divide each term by sin2 x.

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Example 11

y¡=¡csc¡x

y

Use sketching techniques from

y¡=¡sin¡x

Chapter 5 to sketch the graph 1

1 = csc x from sin x the graph of y = sin x for x 2 [¡2¼, 2¼ ]. of y =

x -2p

-p

p

2p

-1

Check your answer using your calculator.

EXERCISE 12H 1 Without using a calculator, find csc x, sec x and cot x for: a sin x = 35 , 0 6 x 6 ¼2 b cos x = 23 , 3¼ 2 < x < 2¼ 2 Without using a calculator, find: ¡ ¢ ¡ ¢ b cot 2¼ a csc ¼3 3

c

sec

¡ 5¼ ¢

d

6

cot (¼)

3 Find the other five trigonometric ratios if: 3 4

a

cos x =

c

sec x = 2 12

e

tan ¯ =

3¼ 2

and

1 2

and 0 < x < and ¼ < ¯
0 and 0 < b < 2¼, find k and b. 17 Use the identity sin A ¡ sin B = 2 cos 12 (A + B) sin 12 (A ¡ B) to show that if sin A = sin B, then either A = B + k2¼ or A + B = ¼ + k2¼: 18

a Prove that cos 3µ = 4 cos3 µ ¡ 3 cos µ by replacing 3µ by (2µ + µ). b Hence, solve the equation 8 cos3 µ ¡ 6 cos µ + 1 = 0 for µ 2 [ ¡¼, ¼ ]:

19

a Write sin 3µ in the form a sin3 µ + b sin µ where a, b 2 Z . b Hence, solve the equation sin 3µ = sin µ for µ 2 [ 0, 3¼ ]:

20 Use the basic definition of periodicity to show algebraically that the period of 2¼ f(x) = sin(nx) is , for all n > 0. n 21

a Write 2 cos x ¡ 5 sin x in the form k cos(x + b) for k > 0, 0 < b < 2¼.

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b Use a to solve the equation 2 cos x ¡ 5 sin x = ¡2 for 0 6 x 6 ¼. 2t 1 ¡ t2 and cos x = . c Given that t = tan( x2 ), prove that sin x = 1 + t2 1 + t2 d Solve 2 cos x ¡ 5 sin x = ¡2 for 0 6 x 6 ¼ using c.

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L

TRIGONOMETRIC EQUATIONS IN QUADRATIC FORM

Sometimes we may be given trigonometric equations in quadratic form. For example, 2 sin2 x + sin x = 0 and 2 cos2 x + cos x ¡ 1 = 0 are clearly quadratic equations where the variables are sin x and cos x respectively. These equations can be factorised and then solved: 2 sin2 x + sin x = 0 sin x(2 sin x + 1) = 0 ) sin x = 0 or ¡ 12 etc.

)

2 cos2 x + cos x + 1 = 0 (2 cos x ¡ 1)(cos x + 1) = 0 ) cos x = 12 or ¡1 etc.

and )

The use of the quadratic formula is often necessary.

EXERCISE 12L 1 Solve for x 2 [ 0, 2¼ ] giving your answers as exact values: b 2 cos2 x = cos x c 2 cos2 x + cos x ¡ 1 = 0 a 2 sin2 x + sin x = 0 d 2 sin2 x + 3 sin x + 1 = 0 g sin 4x = sin 2x

e sin2 x = 2 ¡ cos x f p h sin x + cos x = 2

3 tan x = cot x

2 Solve for x 2 [¡¼, ¼ ] giving your answers as exact values: a 2 sin x + csc x = 3 b sin 2x + cos x ¡ 2 sin x ¡ 1 = 0 4 2 c tan x ¡ 2 tan x ¡ 3 = 0 3 Solve for x 2 [ 0, 2¼ ] : a 2 cos2 x = sin x b

cos 2x + 5 sin x = 0

2 tan2 x + 3 sec2 x = 7

c

M TRIGONOMETRIC SERIES AND PRODUCTS EXERCISE 12M a Simplify 1 + sin x + sin2 x + sin3 x + sin4 x + ::::: + sinn¡1 x .

1

b What is the sum of the infinite series 1 + sin x + sin2 x + sin3 x + ::::: ? c If the series in b has sum 23 , find x such that x 2 [ 0, 2¼ ]. 2 We know that 2 sin x cos x = sin 2x. a Show that: i 2 sin x(cos x + cos 3x) = sin 4x ii

2 sin x(cos x + cos 3x + cos 5x) = sin 6x

b What do you suspect the following would simplify to? i 2 sin x(cos x + cos 3x + cos 5x + cos 7x) ii cos x + cos 3x + cos 5x + ::::: + cos 19x (i.e., 10 terms)

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c Write down the possible generalisation of b ii to n terms.

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3 From sin 2x = 2 sin x cos x we observe that sin x cos x =

sin 2x sin(21 x) = 2 21

a Prove that: sin(23 x) sin(22 x) ii sin x cos x cos 2x cos 4x = i sin x cos x cos 2x = 22 23 b If the pattern observed in a continues, what would: i sin x cos x cos 2x cos 4x cos 8x ii sin x cos x cos 2x:::: cos 32x simplify to? c What is the generalisation of the results in a and b? a Use the principle of mathematical induction to prove that: sin 2nµ cos µ + cos 3µ + cos 5µ + :::::: + cos(2n ¡ 1)µ = , n 2 Z +. 2 sin µ

4

b What does cos µ + cos 3µ + cos 5µ + :::::: + cos 31µ

simplify to?

5 Use the principle of mathematical induction to prove that: sin µ + sin 3µ + sin 5µ + ::::: + sin(2n ¡ 1)µ = for all positive integers n, and hence find the value of 5¼ 9¼ 11¼ 13¼ sin ¼7 + sin 3¼ 7 + sin 7 + sin ¼ + sin 7 + sin 7 + sin 7 .

1 ¡ cos 2nµ 2 sin µ

6 Use the principle of mathematical induction to prove that: sin(2n x) cos x £ cos 2x £ cos 4x £ cos 8x::::::: cos(2n¡1 x) = n 2 £ sin x

for all n 2 Z + .

7 Use the principle of mathematical induction to prove that: · ¸ cos(n + 1)µ sin nµ cos2 µ + cos2 2µ + cos2 3µ + cos2 4µ::::::: cos2 (nµ) = 12 n + sin µ for all n 2 Z + .

REVIEW SET 12A 1 Without using technology draw the graph of y = 4 sin x for 0 6 x 6 2¼: 2 Without using technology draw the graph of y = sin 3x for 0 6 x 6 2¼. ¡ ¢ 3 State the period of: a y = 4 sin x3 b y = ¡2 tan 4x ¢ ¡ 4 Without using technology draw a sketch graph of y = sin x ¡ ¼3 + 2: 5 The table below gives the mean monthly maximum temperature (o C) for Perth Airport in Western Australia. Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Temp 31:5 31:8 29:5 25:4 21:5 18:8 17:7 18:3 20:1 22:4 25:5 28:8

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a A sine function of the form T ¼ A sin B(t¡C)+D is used to model the data. Find good estimates of the constants A, B, C and D without using technology. Use Jan ´ 1, Feb ´ 2, etc. b Check your answer to a using your technology. How well does your model fit?

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6 Use technology to solve for x 2 [ 0, 8 ] : a sin x = 0:382 b

tan

7 Use technology to solve for x 2 [ 0, 8 ] : a sin(x ¡ 2:4) = 0:754 b

¢ ¡ sin x + ¼3 = 0:6049

8 Solve algebraically in terms of ¼: a 2 sin x = ¡1 for x 2 [ 0, 4¼ ]

p 2 sin x ¡ 1 = 0 for x 2 [¡2¼, 2¼]

b

¡x¢ 2

= ¡0:458

9 Solve algebraically in terms of ¼: p a 2 sin 3x + 3 = 0 for x 2 [ 0, 2¼] b sec2 x = tan x + 1 for x 2 [ 0, 2¼] 10 An ecologist studying a species of water beetle estimates the population of a colony over an eight week period. If t is the number of weeks after the initial estimate is made, then the population can ¡ ¢ where be modelled by P (t) = 5 + 2 sin ¼t 3 0 6 t 6 8. a What was the initial population? b What were the smallest and largest populations? c During what time interval(s) did the population exceed 6000?

REVIEW SET 12B 1 Solve algebraically, giving answers in terms of ¼: b 4 sin2 x = 1 a sin2 x ¡ sin x ¡ 2 = 0 a On the same set of axes, sketch y = cos x and y = cos x ¡ 3: ¡ ¢ b On the same set of axes, sketch y = cos x and y = cos x ¡ ¼4 .

2

c On the same set of axes, sketch y = cos x and y = 3 cos 2x: ¡ ¢ d On the same set of axes, sketch y = cos x and y = 2 cos x ¡ ¼3 + 3: 3 In an industrial city, the amount of pollution in the air becomes greater during the working week when factories are operating, and lessens over the weekend. The number of milligrams of pollutants in a cubic metre of air is given by ¡ ¢ 37 P (t) = 40 + 12 sin 2¼ 7 t ¡ 12 where t is the number of days after midnight on Saturday night.

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a What is the minimum level of pollution? b At what time during the week does this minimum level occur?

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4 Find the cosine function represented in the following graphs: y y a b 4

3 2p

x

1 x

8

-4

5 Use technology to solve: a cos x = 0:4379 for 0 6 x 6 10 b cos(x ¡ 2:4) = ¡0:6014 for 0 6 x 6 6 ¡¼¢ b tan 12 6 Find the exact value of: a cos(165o ) 7 Find the exact solutions of: ¢ ¡ a tan x ¡ ¼3 = p13 , x 2 [ 0, 4¼ ]

¡ b cos x +

2¼ 3

¢

= 12 , x 2 [¡2¼, 2¼ ]

8 Find the exact solutions of: p p ¢ ¡ a 2 cos x + ¼4 ¡ 1 = 0, x 2 [ 0, 4¼ ] b tan 2x ¡ 3 = 0, x 2 [ 0, 2¼ ] 9 Simplify:

a

cos3 µ + sin2 µ cos µ

b

d

sin2 µ ¡ 1 cos µ

cos2 µ ¡ 1 sin µ

e

cos2 µ (tan µ + 1)2 ¡ 1

10 Expand and simplify if possible:

(2 sin ® ¡ 1)2

a

5 ¡ 5 sin2 µ

c

(cos ® ¡ sin ®)2

b

REVIEW SET 12C 1 Simplify: 1 ¡ cos2 µ 1 + cos µ

a

sin ® ¡ cos ® sin2 ® ¡ cos2 ®

b

2 Show that: cos µ 1 + sin µ a + = 2 sec µ 1 + sin µ cos µ 3 If sin A =

5 13

and cos A =

4 If sin ® = ¡ 34 , ® 2 [ ¼, sin 2®. 5 If cos x = ¡ 34 6

3¼ 2

12 13

µ b 1+

1 cos µ

c ¶

find the values of:

4 sin2 ® ¡ 4 8 cos ®

¡ ¢ cos µ ¡ cos2 µ = sin2 µ: a

sin 2A

b

cos 2A

] find the value of cos ® and hence the value of

and ¼ < x
j w j ¡ j z j. Once again discuss when the equality case occurs. 3 What transformation moves: a z to z ¤ b z to ¡z

c

z to ¡z ¤

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4 Find the complex number z that satisfies the equation p j z j = 2 10.

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d

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10 50 ¡ = 2 + 9i given z¤ z

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COMPLEX NUMBERS (Chapter 15)

ARGUMENT

³ ´ a b

The direction of the vector

can be described by its angle from the positive real axis.

I

Suppose the complex number z = a + bi is ¡! represented by vector OP as shown alongside. a + bi is the Cartesian form of z: ¡ ! Suppose that OP makes an angle of µ with the positive real axis.

P( a, b) b q

R a

The angle µ is called the argument of z, or simply arg z. arg z = µ has infinitely many possibilities i.e., z 7! arg z = µ not a function. Can you explain why?

is one-to-many and is

To avoid the infinite number of possibilities for µ, we may choose to use µ 2 ] ¡ ¼, ¼ ] which covers one full revolution and guarantees that z 7! arg z = µ is a function. ² ²

Note:

Real numbers have argument of 0 or ¼. Pure imaginary numbers have argument of

¼ 2

or ¡ ¼2 .

POLAR FORM The polar form representation of a complex number is an alternative to Cartesian form, and has many useful applications. r

Any point P which lies on a circle with centre O(0, 0) and radius r, has Cartesian coordinates (r cos µ, r sin µ).

I P(rcos¡q, rsin¡q)

So, z = r cos µ + ir sin µ = r(cos µ + i sin µ)

q

-r

r

R

But r = j z j and if we define cis µ = cos µ + i sin µ

-r

then z = j z j cis µ. Consequently: Any complex number z has Cartesian form z = x + yi or polar form z = j z j cis µ where j z j is the modulus of z, µ is the argument of z, and cis µ = cos µ + i sin µ. We will soon see that polar form is extremely powerful for dealing with multiplication and division of complex numbers, as well as quickly finding powers and roots of numbers (see De Moivre’s theorem). A useful identity is:

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If z = j z j cis µ, then z ¤ = j z j cis (¡µ).

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EULER FORM where r = j z j and µ = arg z.

z can also be written in Euler form as z = reiµ cis µ = eiµ .

In other words,

We will derive this identity in Chapter 27.

For example, consider the complex number z = 1 + i: p j z j = 2 and µ = ¼4 , p ¡ ¢ p ¼ ) 1 + i = 2 cis ¼4 = 2ei 4 p ¡ ¢ is the polar form of 1 + i So, 2 cis ¼4 p i¼ and 2e 4 is the Euler form of 1 + i.

(1,¡1)

I

1 p 4

R 1

Example 9 b ¡3

a 2i

Write in polar form: a

b

c

I

2

1¡i

c

I

I

p

p 2

-3

R

- p4

R -i

j 2i j = 2

j¡3 j = 3

¼ 2

µ=¼

µ=

) 2i = 2 cis

j1 ¡ ij =

R 1-i

p p 1+1= 2

µ = ¡ ¼4 p ¡ ¢ ) 1 ¡ i = 2 cis ¡ ¼4

) ¡3 = 3 cis ¼

¼ 2

1

Example 10 p ¡ ¢ Convert 3 cis 5¼ 6 to Cartesian form.

(

- 3 2

I

, 12 )

5p 6

1

R

p ¡ ¢ 3 cis 5¼ 6 p £ ¡ 5¼ ¢ ¡ ¢¤ = 3 cos 6 + i sin 5¼ 6 i p h p3 1 = 3 ¡ 2 +i£ 2 = ¡ 32 +

p 3 2 i

EXERCISE 15B.3

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1 Find the modulus and argument of the following complex numbers and hence write them in polar form: a 4 b 2i c ¡6 d ¡3i p p e 1+i f 2 ¡ 2i g ¡ 3+i h 2 3 + 2i

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2 What complex number cannot be written in polar form? Why? 3 Convert k + ki to polar form. (Careful! You must consider k > 0, k = 0, k < 0.) 4 Convert to Cartesian form without using a calculator: ¡ ¢ ¡ ¢ b 8 cis ¼4 a 2 cis ¼2 p p ¡ ¢ ¡ ¢ e 2 cis ¡ ¼4 3 cis 2¼ d 3 5

¡¼¢

c

4 cis

f

5 cis ¼

6

a Find the value of cis 0. b Find the modulus of cis µ, i.e., j cis µ j : c Show that cis ® cis ¯ = cis (® + ¯).

MULTIPLYING AND DIVIDING IN POLAR FORM cis µ has three useful properties. These are: ²

cis µ £ cis Á = cis (µ + Á) cis µ = cis (µ ¡ Á) cis Á cis (µ + k2¼) = cis µ for all k 2 Z .

² ²

The first two of these are similar to index laws: aµ aÁ = aµ+Á Proof: ²

and

aµ = aµ¡Á . aÁ

cis µ £ cis Á = (cos µ + i sin µ)(cos Á + i sin Á) = [cos µ cos Á ¡ sin µ sin Á] + i[sin µ cos Á + cos µ sin Á] = cos(µ + Á) + i sin(µ + Á) fcompound angle identitiesg = cis (µ + Á) cis µ cis (¡Á) cis µ = £ cis Á cis Á cis (¡Á)

²

=

²

cis (µ + k2¼) = cis µ £ cis (k2¼) = cis µ £ 1 = cis µ

cis (µ ¡ Á) cis 0

= cis (µ ¡ Á) fas cis 0 = 1g

(1,¡0)

The above results are even easier to prove if we use Euler’s form. Using cis µ = eiµ and cis Á = eiÁ , we find:

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= ei(µ+Á) = cis (µ + Á)

cis µ eiµ = iÁ cis Á e = eiµ¡iÁ = ei(µ¡Á) = cis (µ ¡ Á)

0

²

cis µ cis Á = eiµ eiÁ

5

²

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Example 11 a cis

Use the properties of cis to simplify:

a

cis = cis = cis = cis = cos

¡¼¢ 5

¡¼

cis

5 + ¡ 5¼ ¢

¡ 3¼ ¢ ¢

10

¼ 2 ¼ 2

5

cis

+ i sin ¼2

= 0 + i(¡1) = ¡i

= 0 + i(1) =i

Example 12

¡ 3¼ ¢ 10

¡ ¢ cis ¼5 ¡ ¢ cis 7¼ 10

b

¡ ¢ cis ¼5 ¡ 7¼ ¢ cis 10 ¢ ¡ = cis ¼5 ¡ 7¼ 10 ¡ ¢ = cis ¡ ¼2 ¡ ¢ ¡ ¢ = cos ¡ ¼2 + i sin ¡ ¼2

b

10

3¼ 10

¡¼¢

I (0,¡1) p 2

R

-p 2

(0,-1)

(see Example 8 earlier)

What transformation moves z to iz?

Let z = r cis µ and use i = 1 cis ) iz = r cis µ £ cis ¼2 ¢ ¡ = r cis µ + ¼2

¼ 2.

So, z has been rotated anti-clockwise by

Example 13

Simplify cis

¡ 107¼ ¢ 6

= 17 56 ¼ = 18¼ ¡ ¼6 ¢ ¡ ¢ ¡ = cis 18¼ ¡ ¼6 ) cis 107¼ 6 ¡ ¢ fcis (µ + k2¼) = cis µg = cis ¡ ¼6 =

about O.

:

107¼ 6

p 3 2

¼ 2

I -p 6

¡ 12 i

R

(

3 2

, -21)

Example 14 p ¡ ¢ Write z = 1 + 3i in polar form and then multiply it by 2 cis ¼6 . Illustrate what has happened on an Argand diagram. ¡ ¢ What transformations have taken place when multiplying by 2 cis ¼6 ?

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q p p If z = 1 + 3i, then j z j = 12 + ( 3)2 ³ p ´ =2 ) z = 2 12 + 23 i ¡ ¢¢ ¡ ¡ ¢ ) z = 2 cos ¼3 + i sin ¼3 ¡ ¢ ) z = 2 cis ¼3

I

5

a

100

a b c

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(1 +

p 3i) £ 2 cis

¼ 6

¡¼¢

£ 2 cis ¡ ¼3 ¼ ¢ + ¡ ¼3 ¢ 6 2

= 2 cis = 4 cis = 4 cis

¡¼¢ 6

= 4(0 + 1i) = 4i ¡ ¢ c When z was multiplied by 2 cis ¼6 its modulus (length) was doubled and it was rotated through ¼6 .

I

b

(0,¡4) p 6

4i

1+~`3\\i

p 3

R

If a complex number z is multiplied by r cis µ then its modulus is multiplied by r and its argument is increased by µ.

Example 15 Use complex numbers to write cos cos = cis = cis

¡ 7¼ ¢ 12

¡ 7¼ ¢

+ i sin

12

and sin

12

¡ 7¼ ¢

in simplest surd form.

12

¡ 7¼ ¢

I

12

+

4¼ 12

(, ) ( 1 2

1

12

¡ 3¼

¡ 7¼ ¢

¢

3 2

1 2

¡ ¢ ¡ ¢ = cis ¼4 £ cis ¼3 fcis (µ + Á) = cis µ £ cis Ág ³ ´³ p ´ = p12 + p12 i 12 + 23 i ³p ´ ³ p ´ 1 1 p3 p3 + p ¡ + i = 2p 2 2 2 2 2 2 2 Equating real parts: cos 7¼ 12 =

³

p ´ 1¡p 3 2 2

Equating imaginary parts: sin 7¼ 12 =

£

p p2 2

=

p 3+1 p 2 2

=

p p 6+ 2 4

p 3

p 4

1

,

1 2

) R

p p 2¡ 6 4

EXERCISE 15B.4

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[cis µ]3 ¡ ¢ ¡ 8¼ ¢ 2 cis 2¼ 5 £ 4 cis 5

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£p ¡ ¢¤ 4 2 cis ¼8

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1 Use the properties of cis to simplify: cis 3µ a cis µ cis 2µ b cis µ ¡¼¢ ¡¼¢ ¡ ¢ ¡¼¢ e 2 cis 12 cis ¼6 d cis 18 cis 9 p ¡¼¢ ¡ ¢ 4 cis 12 32 cis ¼8 ¡ ¢ ¢ ¡ h p g 2 cis 7¼ 2 cis ¡ 7¼ 12 8

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2 Use the property cis (µ + k2¼) = cis µ a

b

cis 17¼

to evaluate: c

cis (¡37¼)

cis

¡ 91¼ ¢ 3

3 If z = 2 cis µ: a What is j z j and arg z? b Write z ¤ in polar form. c Write ¡z in polar form. Note: ¡2 cis µ is not in polar form as the coefficient of cis µ is a length and so must be positive. d Write ¡z ¤ in polar form. a b c d

4

Write i in polar form. z = r cis µ is any complex number. Write iz in polar form. Explain why iz is the anti-clockwise rotation of z about O through ¼2 radians. What transformation maps z onto ¡iz? Give reasoning in polar form.

5 Write in polar form: a cos µ ¡ i sin µ b Use a above to complete this sentence: If z = r cis µ then z ¤ = ::::: in polar form.

sin µ ¡ i cos µ

6 Use complex number methods to find, in simplest surd form: ¡¼¢ ¡¼¢ ¡ ¢ ¡ ¢ a cos 12 and sin 12 b cos 11¼ and sin 11¼ 12 12

PROPERTIES OF ARGUMENT ²

The basic properties of argument are:

²

arg(zw) = arg z + arg w

²

n

arg

³z´ w

= arg z ¡ arg w

arg (z ) = n arg z

Notice that they are identical to the laws of logarithms, with arg replaced by log or ln. Properties of modulus and argument can be proved jointly using polar form.

Example 16 Use polar form to establish that j zw j = j z j j w j and arg(zw) = arg z + arg w. Let z = j z j cis µ

and w = j w j cis Á.

Now zw = j z j cis µ £ j w j cis Á = j z j j w j cis (µ + Á) fproperty of cisg | {z } non-negative )

j zw j = j z j j w j

fthe non-negative number multiplied by cis (.....) g

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and arg(zw) = µ + Á = arg z + arg w.

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Example 17 If z = a

p 2 cis µ, find the modulus and argument of: a 2z

p 2z = 2 2 cis µ

b

b iz

p j 2z j = 2 2 and arg (2z) = µ

)

¼ 2

i = cis

p iz = cis ¼2 £ 2 cis µ p p ¡ ¢ = 2 cis ¼2 + µ So, j iz j = 2 and arg (iz) =

)

c

c (1 ¡ i)z

¼ 2

+µ

p ¡ ¢ 2 cis ¡ ¼4 p ¡ ¢ p (1 ¡ i)z = 2 cis ¡ ¼4 £ 2 cis µ ¡ ¢ = 2 cis ¡ ¼4 + µ 1¡i =

I 1 -p 4

)

R 1

-i

)

j (1 ¡ i)z j = 2 and arg ((1 ¡ i) z) = µ ¡

¼ 4

Example 18 Suppose z = cis Á where Á is acute. Find the modulus and argument of z + 1: jzj = 1 ) z lies on the unit circle ¡! z + 1 is OB ffound vectoriallyg

I A

1

z

B

z+1 R

C

1

OABC is a rhombus ) arg(z + 1) =

A M

z

O

fdiagonals bisect the angles of the rhombusg ¡ ¢ OM Also cos µ2 = 1 ¡ ¢ ) OM = cos µ2 ¡ ¢ ) OB = 2 cos µ2 ¡ ¢ ) j z + 1 j = 2 cos µ2

B

1

q 2

1

µ 2

C

EXERCISE 15B.5 1 Use polar form to establish: ¯z ¯ ³z´ jzj ¯ ¯ and arg = arg z ¡ arg w, provided w 6= 0: ¯ ¯= w jwj w

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2 Suppose z = 3 cis µ. Determine the modulus and argument of: c iz d a ¡z b z¤

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a If z = cis Á where Á is acute, determine the modulus and argument of z ¡ 1. b Using a, write z ¡ 1 in polar form.

3

c Hence write (z ¡ 1)¤

in polar form.

¡! ¡! 4 ABC is an equilateral triangle. Suppose z1 represents OA, z2 represents OB and z3 ¡! represents OC. a Explain what vectors represent z2 ¡ z1 and z3 ¡ z2 . ¯ ¯ ¯ z2 ¡ z1 ¯ ¯ ¯: b Find ¯ z3 ¡ z2 ¯ ¶ µ z2 ¡ z1 : c Determine arg z3 ¡ z2 µ ¶3 z2 ¡ z1 : d Use b and c to find the value of z3 ¡ z2

B

I

A C R

5 Given that z = a ¡ i where a is real, find the exact value of a if arg z = ¡ 5¼ 6 . i¼

to find the values of ei¼ and e 2 . cis µ b Prove that cis µ cis Á = cis (µ + Á) and = cis (µ ¡ Á). cis Á p c If z = cis µ, find the argument of i z ii iz iii ¡iz 2 iv a Use eiµ = cos µ + i sin µ

6

i z

FURTHER CONVERSION BETWEEN CARTESIAN AND POLAR FORMS I

r| cis {z µ} then: Polar form y tan µ = , x y x , sin µ = p cos µ = p 2 2 2 x +y x + y2

If z = x + iy = | {z } Cartesian form p r = x2 + y2 ,

P(x, y) r q

y R

x

POLAR TO CARTESIAN ¡ ¢ z = 2 cis ¼8 ¢ ¡ = 2 cos ¼8 + i sin ¼8

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¼ 1:85 + 0:765i

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CARTESIAN TO POLAR p p ¡3 1 z = ¡3+i has r = (¡3)2 + 12 = 10 and cos µ = p , sin µ = p 10 10 | {z } I ³ ´ ) µ = ¼ ¡ sin¡1 p110 fquadrant 2g 1 q ³ ´ R ) (or cos¡1 p¡3 3 10

)

¡3 + i ¼

p 10 cis (2:82) TI

Alternatively (TI-83):

C

MATH CPX takes us to the complex number menu.

Pressing 5 brings up abs( for calculating the modulus (absolute value). To find the modulus of ¡3 + i, press

+

3

2nd i

) ENTER ~`1`0

MATH CPX then 4 brings up angle( for calculating the

argument. To find the argument of ¡3 + i, press 3

+

) ENTER

2nd i

EXERCISE 15B.6 1 Use your calculator to convert to Cartesian form: p p ¢ ¡ a 3 cis (2:5187) b 11 cis ¡ 3¼ 8

c

2:836 49 cis (¡2:684 32)

2 Use your calculator to convert to polar form: a 3 ¡ 4i b ¡5 ¡ 12i

c

¡11:6814 + 13:2697i

3 Add the following using a + bi surd form and convert your answer to polar form: ¡ ¢ ¡ ¢ ¡ ¡2¼ ¢ ¡ ¢ b 2 cis 2¼ a 3 cis ¼4 + cis ¡3¼ 4 3 + 5 cis 3 4 Use the sum and product of roots to find the real quadratic equations with roots of: p ¡ 4¼ ¢ ¡ ¢ ¡ ¢ p ¡ ¢ b a 2 cis 2¼ 2 cis ¼4 , 2 cis ¡¼ 3 , 2 cis 3 4

C

DE MOIVRE’S THEOREM

Polar form enables us to easily calculate powers of complex numbers. Notice that if z = j z j cis µ then z 2 = j z j cis µ £ j z j cis µ

z3 = z2 z

and

2

= j z j2 cis 2µ £ j z j cis µ

2

= j z j cis (2µ + µ)

= j z j cis (µ + µ)

3

= j z j cis 2µ

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= j z j3 cis 3µ

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The generalisation of this process is De Moivre’s Theorem: (j z j cis µ)n = j z jn cis nµ Proof for n 2 Z + :

(using mathematical induction)

Pn is: (j z j cis µ)n = j z jn cis nµ (1) If n = 1, then (j z j cis µ)1 = j z j cis µ

) P1 is true.

k

(2) If Pk is true, then (j z j cis µ)k = j z j cis kµ Thus (j z j cis µ)k+1 = (j z j cis µ)k £ j z j cis µ k

= j z j cis kµ £ j z j cis µ k+1

cis (kµ + µ)

k+1

cis (k + 1) µ

= jzj

= jzj

findex lawg fusing Pk g findex law and cis propertyg

Thus Pk+1 is true whenever Pk is true and P1 is true. ) Pn is true fPrinciple of mathematical inductiong cis (¡nµ) = cis (0 ¡ nµ) =

We observe also that

1 cis nµ 1 = [ cis µ ]n

) cis (¡nµ) =

cis 0 cis nµ

fas

cis µ = cis (µ ¡ Á)g cis Á

fas cis 0 = 1g ffor n a positive integerg

= [ cis µ ]¡n Also [cis

¡µ¢ n ¡ ¡ µ ¢¢ = cis µ n ] = cis n n

so the theorem is true for all n 2 Z ¡ ¢ 1 and so [cis µ] n = cis nµ

So, De Moivre’s theorem seems to hold for any integer n and for

1 n.

DE MOIVRE’S THEOREM (j z j cis µ)n = j z jn cis nµ

for all rational n.

Example 19 p Find the exact value of ( 3 + i)8 Check your answer by calculator.

using De Moivre’s theorem.

qp p p 3 + i has modulus ( 3)2 + 12 = 4 = 2 ³p ´ p 3 + i = 2 23 + 12 i ) = 2 cis

I 1 R

q

¼ 6

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p ¡ ) ( 3 + i)8 = 2 cis

¢ ¼ 8 6

8

¡ 8¼ ¢

8

¡ 4¼ ¢

= 2 cis

I

6

= 2 cis 3 ³ = 28 ¡ 12 ¡

4p 3

R

p ´ 3 2 i

(

-1 - 3 2 2

p = ¡128 ¡ 128 3i

,

)

Example 20 By considering cos 2µ + i sin 2µ, deduce the double angle formulae for cos 2µ and sin 2µ.

Now

cos 2µ + i sin 2µ = cis 2µ fDe Moivre’s theoremg = [cis µ]2 = [cos µ + i sin µ]2 = [cos2 µ ¡ sin2 µ] + i[2 sin µ cos µ]

Equating imaginary parts, Equating real parts,

sin 2µ = 2 sin µ cos µ cos 2µ = cos2 µ ¡ sin2 µ

EXERCISE 15C 1 Use De Moivre’s theorem to simplify: ¢10 ¡p ¡ a 2 cis ¼5 b cis d

p 5 cis

¼ 7

¢ ¼ 36 12

p 3 8 cis

e

¼ 2

2 Use De Moivre’s theorem to find the exact value of: p a (1 + i)15 b (1 ¡ i 3)11 p 1 d (¡1 + i)¡11 e ( 3 ¡ i) 2

c

¡p 2 cis

¢ ¼ 12 8

f

¡ 8 cis

¢ 53

c f

¼ 5

p p ( 2 ¡ i 2)¡19 p 5 (2 + 2i 3)¡ 2

3 Use your calculator to check the answers to 2. a Suppose z = j z j cis µ where ¡¼ < µ 6 ¼. p Use De Moivre’s theorem to find z in terms of j z j and µ. p b What restrictions apply to Á = arg( z)? p c True or false? “ z has a non-negative real part.”

4

5 Use De Moivre’s theorem to explain why j z n j = j z jn

and arg (z n ) = n arg z:

6 Show that cos µ ¡ i sin µ = cis (¡µ). Hence, simplify (cos µ ¡ i sin µ)¡3 .

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7 Write z = 1 + i in polar form and hence write z n in polar form. Find all values of n for which: a z n is real b z n is purely imaginary.

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8 If j z j = 2 and arg z = µ, determine the modulus and argument of: 1 i a z3 b iz 2 c d ¡ 2 z z z2 ¡ 1 9 If z = cis µ, prove that = i tan µ. z2 + 1 10

a Use complex number methods to deduce that: ii sin 3µ = 3 sin µ ¡ 4 sin3 µ i cos 3µ = 4 cos3 µ ¡ 3 cos µ b Hence, find tan 3µ in terms of tan µ only. c Using a and b above, solve the equations: 4x3 ¡ 3x = ¡ p12

i

ii

p p x3 ¡ 3 3x2 ¡ 3x + 3 = 0:

11 Points A, B and C form an isosceles triangle with a right angle at B. Let the points A, B and C be represented by the complex numbers z1 , z2 , and z3 respectively. a Show that (z1 ¡ z2 )2 = ¡(z3 ¡ z2 )2 . b If ABCD forms a square, what complex number represents the point D? Give your answer in terms of z1 , z2 and z3 . 12 Find a formula for

a b

B A

cos 4µ in terms of cos µ sin 4µ in terms of cos µ and sin µ

prove that z n +

a If z = cis µ

13

C

1 = 2 cos nµ: zn

1 = 2 cos µ. z 1 c Use the binomial theorem to expand (z + )3 , and simplify your result. z 3 d By using a, b and c above show that cos µ = 14 cos 3µ + 34 cos µ. p p ¡ ¢ 6 is ¡5 2¡3 e Hence show the exact value of cos3 13¼ . Hint: 13¼ 12 16 12 =

b Hence, explain why z +

then z n ¡

14 Show that if z = cis µ that sin3 µ =

3 4

sin µ ¡

1 4

1 = 2i sin nµ zn

+ ¼3

and hence

sin 3µ?

15 Use the results of 13 and 14 to prove that sin3 µ cos3 µ =

D

3¼ 4

1 32 (3 sin 2µ

¡ sin 6µ).

ROOTS OF COMPLEX NUMBERS

SOLVING zn¡=¡c We will examine solutions of equations of the form z n = c where n is a positive integer and c is a complex number.

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The nth roots of complex number c are the n solutions of z n = c.

Definition:

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For example, the 4th roots of 2i are the four solutions of z 4 = 2i. The roots may be found by factorisation, but this is sometimes difficult. It is therefore desirable to have an alternative method such as the nth roots method presented in the following example.

Example 21 a factorisation

Find the four 4th roots of 1 by:

b the ‘nth roots method’.

We need to find the 4 solutions of z 4 = 1. I R b By the ‘nth roots method’, a By factorisation, z4 = 1 1 ) z4 ¡ 1 = 0 z4 = 1 2 2 ) (z + 1)(z ¡ 1) = 0 ) z 4 = 1 cis (0 + k2¼) fpolar formg 1 (z + i)(z ¡ i)(z + 1)(z ¡ 1) = 0 ) z = [cis (k2¼)] 4 ¢ ¡ ) z = §i or §1 fDe Moivreg ) z = cis k2¼ 4 ¡ k¼ ¢ ) z = cis 2 ) z = cis 0, cis ¼2 , cis ¼, cis fletting k = 0, 1, 2, 3g ) z = 1, i, ¡1, ¡i Note: ²

3¼ 2

The factorisation method is fine provided the polynomial factorises easily. This is not usually the case. The substitution of k = 0, 1, 2, 3 to find the 4 roots could be done using any 4 consecutive integers for k. Why?

²

EXERCISE 15D.1 1 Find the three cube roots of 1 using: 2 Solve for z:

z 3 = ¡8i

a

a

b

factorisation

the ‘nth roots method’.

z 3 = ¡27i

b

3 Find the three cube roots of ¡1, and display them on an Argand diagram. 4 Solve for z:

z 4 = 16

a

z 4 = ¡16

b

5 Find the four fourth roots of ¡i, and display them on an Argand diagram. 6 Solve the following and display the roots on an Argand diagram: a d

z 3 = 2 + 2i p z4 = 3 + i

b

z 3 = ¡2 + 2i

c

e

z 5 = ¡4 ¡ 4i

f

1 2

p

+ 23 i p z 3 = ¡2 3 ¡ 2i z2 =

Example 22

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Find the fourth roots of ¡4 in the form a + bi and then factorise z 4 + 4 into linear factors. Hence, write z 4 + 4 as a product of real quadratic factors.

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The fourth roots of ¡4 are solutions of z 4 = ¡4 ) z 4 = 4 cis (¼ + k2¼)

I p

)

z = [4 cis (¼ + k2¼)] µ ¶ ¼ + k2¼ 1 z = 4 4 cis 4

)

1

R

-4

1 4

1

1

1

)

5¼ 7¼ 2 2 z = 2 2 cis ¼4 , 2 2 cis 3¼ fletting k = 0, 1, 2, 3g 4 , 2 cis 4 , 2 cis 4 ´ p ³ ´ p ³ ´ p ³ ´ p ³ 1 z = 2 p2 + p12 i , 2 ¡ p12 + p12 i , 2 ¡ p12 ¡ p12 i , 2 p12 ¡ p12 i

)

z = 1 + i, ¡1 + i, ¡1 ¡ i, 1 ¡ i

)

Roots 1 § i have sum = 2 and product = (1 + i)(1 ¡ i) = 2 and ) come from the quadratic factor z 2 ¡ 2z + 2. Roots ¡1 § i have sum = ¡2 and product = (¡1 + i)(¡1 ¡ i) = 2 and ) come from the quadratic factor z 2 + 2z + 2.

In examples such as this we observe the connection between polynomial methods and complex number theory.

Thus z 4 + 4 = (z 2 ¡ 2z + 2)(z 2 + 2z + 2) 7 Find the four solutions of z4¡+¡1¡=¡0 giving each in the form a¡+¡bi, and display them on an Argand diagram. Hence write z4¡+¡1 as the product of two real quadratic factors. ³p ´2 3 1 ¡ i 2 2 8 Consider z = ¡ ¢ ¡ ¢ ¼ ¼ 5 ¼ ¼ 25 cos 10 ¡ i sin 10 cos 30 + i sin 30 a Using polar form and De Moivre’s theorem, find the modulus and argument of z. b Hence show that z is a cube root of 1. c By simplifying and without using a calculator, show that (1 ¡ 2z)(2z 2 ¡ 1) is a real number. a Write ¡16i in polar form. b The fourth root of ¡16i which lies in the second quadrant is denoted by z. Express z exactly in: i polar form ii Cartesian form.

9

SUMMARY OF SOLUTIONS OF zn = c (nth roots of c) ² ² ²

There are exactly n nth roots of c. If c 2 R , the complex roots must occur in conjugate pairs. If c 2 = R , the complex roots do not all occur in conjugate pairs.

²

The roots of z n will all have the same modulus which is j c j n . Thus on an Argand diagram, the roots will all be the same distance from the origin

1

1

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and hence lie on a circle radius j c j n .

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COMPLEX NUMBERS (Chapter 15) 1

²

The roots on the circle r = j c j n will be equally spaced around the circle. If you join all the points you will get a geometric shape that is a regular polygon. For example, n = 3 (equilateral ¢), n = 4 (square) n = 5 (regular pentagon) n = 6 (regular hexagon) etc.

THE nth ROOTS OF UNITY The nth roots of unity are the solutions of z n = 1.

Example 23 Find the three cube roots of unity and display them on an Argand diagram. If w is the root with smallest positive argument, show that the roots are 1, w and w2 and that 1 + w + w2 = 0. The cube roots of unity are the solutions of z 3 = 1. But 1 = cis 0 = cis (0 + k2¼) for all k 2 Z ) z 3 = cis (k2¼) 1

) )

z = [cis (k2¼)] 3 fDe Moivre’s theoremg ¡ k2¼ ¢ z = cis 3 ¡ 4¼ ¢ ¡ ¢ fletting k = 0, 1, 2g z = cis 0, cis 2¼ 3 , cis 3

)

z = 1, ¡ 12 +

)

¡ 12 ¡

p 3 2 i

(

-1 2

¡ 2¼ ¢ 2 ¡ 4¼ ¢ 3 ] = cis 3 ¡ ¢ 2 ) the roots are 1, w and w where w = cis 2¼ 3 ³ ³ p ´ p ´ and 1 + w + w2 = 1 + ¡ 12 + 23 i + ¡ 12 ¡ 23 i = 0 w = cis

¡ 2¼ ¢

p 3 2 i,

,

3 2

)

I

and w2 = [cis

3

R 1

(

-1 - 3 2 2

,

)

EXERCISE 15D.2 1 In Example 23 we showed that the cube roots of 1 are 1, w, w2 where w = cis

¡ 2¼ ¢ 3 .

a Use this fact to solve the following equations, giving your answers in terms of w: i (z + 3)3 = 1 ii (z ¡ 1)3 = 8 iii (2z ¡ 1)3 = ¡1 b Show by vector addition that 1 + w + w2 = 0 if 1, w and w2 are the cube roots of unity. 2 In Example 21 we showed that the four fourth roots of unity were 1, i, ¡1, ¡i. a Is it true that the four fourth roots of unity can be written in the form 1, w, w2 , w3 where w = cis ¼2 ?

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b Show that 1 + w + w2 + w3 = 0. c Show by vector addition that b is true.

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3

a Find the 5 fifth roots of unity and display them on an Argand diagram. b If w is the root with smallest positive argument, show that the roots are 1, w, w2 , w3 and w4 . c Simplify (1+w+w2 +w3 +w4 )(1¡w) and hence show that 1+w+w2 +w3 +w4 must be zero. d Show by vector addition that 1 + w + w2 + w3 + w4 = 0. ¡ ¢ is the nth root of unity with the smallest positive argument, show 4 If w = cis 2¼ n that: a the n roots of z n = 1 are 1, w, w2 , w3 , ....., wn¡1 b 1 + w + w2 + w3 + :::::: + wn¡1 = 0 Note: The roots of unity lie on the unit circle, equally spaced around the unit circle.

5 Show that for any complex number ®, the sum of the n zeros of z n = ® is 0.

E FURTHER COMPLEX NUMBER PROBLEMS The following questions combine complex number theory with topics covered in previous chapters.

EXERCISE 15E 1 Solve for z: z 2 ¡ (2 + i)z + (3 + i) = 0 2 Find the Cartesian equation for the locus of P(x, y) if z = x + iy and b arg(z ¡ i) = ¼6 c jz + 3j + jz ¡ 3j = 8 a z ¤ = ¡iz 3 Use the binomial expansion of (1 + i)2n to prove that: ¡2n¢ ¡2n¢ ¡2n¢ ¡2n¢ ¡ ¢ ¡ n¼ ¢ n 2n n + 0 ¡ 2 + 4 ¡ 6 + ...... +(¡1) 2n = 2 cos 2 , n 2 Z : 4 By considering 1 + cis µ + cis 2µ + cis 3µ + ...... + cis nµ n X find cos rµ. r=0

5 Prove that 1 + cis µ = 2 cos n X ¡n¢ r cos(rµ).

¡µ¢ 2

cis

¡µ¢ 2

as a geometric series,

and hence determine the sum of the series

r=0

REVIEW SET 15A 1 Find the real and imaginary parts of (i ¡

p 5 3) .

2 If z = x+yi and P(x, y) moves in the complex plane, find the Cartesian equation for: a jz ¡ ij = jz + 1 + ij b z ¤ ¡ iz = 0

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3 Find j z j if z is a complex number and j z + 16 j = 4 j z + 1 j.

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4 Points A and B arepthe representations in the complex plane of the numbers z = 2¡2i and w = ¡1 ¡ 3i respectively. a Given that the origin is O, find the angle AOB in terms of ¼. b Calculate the argument of zw in terms of ¼. p c k ¡ ki where k < 0 5 Write in polar form: a ¡5i b 2 ¡ 2i 3 6 Given that z = (1 + bi)2 arg z = ¼3 .

where b is real and positive, find the exact value of b if

a Prove that cis µ £ cis Á = cis (µ + Á). p b If z = 2 2 cis ®, write (1 ¡ i)z in polar form. Hence find arg[(1 ¡ i)z].

7

¡! ¡! 8 z1 ´ OA and z2 ´ OB represent two sides of a right angled isosceles triangle OAB. z2 a Determine the modulus and argument of 12 . A z2 z1 b Hence, deduce that z12 + z22 = 0. p ¡ ¢ ¢ p ¡ and w = 4 b cos ¼4 ¡ i sin ¼4 . 9 Let z = 4 a cos ¼6 + i sin ¼6

B

z2 O

³ z ´4 Find in terms of a and b the exact values of the real and imaginary parts of . w ¡ 2¼ ¢ 10 a List the five fifth roots of 1 in terms of w where w = cis 5 . b Display the roots in a on an Argand diagram. c By considering the factorisation of z 5 ¡ 1 in two different ways, show that: z 4 + z 3 + z 2 + z + 1 = (z ¡ w)(z ¡ w2 )(z ¡ w3 )(z ¡ w4 ). d Hence, find the value of (2 ¡ w)(2 ¡ w2 )(2 ¡ w3 )(2 ¡ w4 ). 11 Find the cube roots of ¡8i, giving your answers in the form a + bi where a and b do not involve trigonometric ratios.

REVIEW SET 15B 1 Let z1 = cos ¼6 + i sin ¼6 and z2 = cos ¼4 + i sin ¼4 . µ ¶3 z1 in the form z = a + bi. Express z2 2 If z = 4 + i and w = 2 ¡ 3i, find: b j w ¡ z¤ j a 2w¤ ¡ iz

¯ 10 ¯ ¯z ¯

c

d

arg(w ¡ z)

2 ¡ 3i = 3 + 2i. 2a + bi

3 Find rationals a and b such that

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4 If z = x + yi and P(x, y) moves in the complex plane, find the Cartesian equation for the curve: ¯ ¯ ¯ z+2 ¯ ¯=2 b ¯¯ a arg(z ¡ i) = ¼2 z¡2 ¯ p p 5 Write 2 ¡ 2 3i in polar form. Hence find all values of n for which (2 ¡ 2 3i)n is real.

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6 Determine the cube roots of ¡27. 7 If z = 4 cis µ, find the modulus and argument of: 1 b a z3 c iz ¤ z 8 Prove the following: a arg(z n ) = n arg z for all complex numbers z and rational n. ³ z ´¤ z¤ = ¤ for all z and for all w 6= 0: b w w 9 Find n such that each of the following can be written in the form [cis µ]n : 1 a cos 3µ + i sin 3µ b c cos µ ¡ i sin µ cos 2µ + i sin 2µ 10 Determine the fifth roots of 2 + 2i. 1 is real, prove that either j z j = 1 or z is real. z 12 If z = cis µ, prove that: 1 a jzj = 1 b z¤ = c sin4 µ = 18 (cos 4µ ¡ 4 cos 2µ + 3) z

11 If z +

13 If w is the root of z 5 = 1 with smallest positive argument, find real quadratic equations with roots of: b w + w4 and w2 + w3 a w and w4 14 If j z + w j = j z ¡ w j prove that arg z and arg w differ by

¼ 2.

15 The complex number z is a root of the equation j z j = j z + 4 j. a Show that the real part of z is ¡2. b Let v and w be two possible values of z such that j z j = 4. i On an Argand diagram, sketch the points that represent v and w, given that v is in the 2nd quadrant. ii Show that arg v = 2¼ 3 . iii Find arg w where ¡¼ < arg w 6 ¼. µ m ¶ v w in terms of m and ¼. i Find arg c i vmw ii Hence find a value of m for which is a real number. i

REVIEW SET 15C 1 What single transformation maps z onto:

a

z¤

b

¡z

c

iz?

2 z and w are non-real complex numbers with the property that both z + w and zw are real. Prove that z ¤ = w.

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3 If (x + iy)n = X + Y i where n is a positive integer, show that X 2 + Y 2 = (x2 + y2 )n .

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³ ´ 4 Prove that j z ¡ w j2 + j z + w j2 = 2 j z j2 + j w j2 . 5 z 5 = 1 has roots 1, ®, ®2 , ®3 and ®4 where ® = cis

¡ 2¼ ¢ 5

:

a Prove that 1 + ® + ®2 + ®3 + ®4 = 0. µ ¶5 z+2 = 1 in terms of ®. b Solve the equation z¡1 ¯ ¯ ¯ z+1 ¯ ¯ ¯ = 1, prove that z is purely imaginary. 6 If z 6= 0 and ¯ z¡1 ¯ 7 If z = cis Á and w =

1+z , show that w = cis Á also. 1 + z¤

p 8 Write ¡1 + i 3 in polar form and hence find the values of m for which p (¡1 + i 3)m is real. ³ ´ ³ ´ 9 Prove that cis µ + cis Á = 2 cos µ¡Á cis µ+Á and hence show that 2 2 µ ¶5 ¡ ¢ z+1 for n = 1, 2, 3 and 4. = 1 has solutions of the form z = i cot n¼ 5 z¡1 10 Find the cube roots of ¡64i, giving your answers in the form a + bi where a and b are real. a (2z)¡1

11 If z = cis µ, find the modulus and argument of:

b 1¡z

12 Illustrate the region defined by fz: 2 6 j z j 6 5 and ¡ ¼4 < arg z 6 ¼2 g. Show clearly all included boundary points. ¯ ¯ µ ¶ ¯1¯ 1 1 ¯ ¯ for z 6= 0 and arg = ¡ arg z: 13 Use polar form to deduce that ¯ ¯ = z jzj z 14 If z = cis ®, write 1 + z in polar form and hence determine the modulus and argument of 1 + z. ¡ ¢ ¡ ¢ Hint: sin µ = 2 sin µ2 cos µ2 15 ¢P1 P2 P3 is an equilateral triangle, as illustrated. ¡¡! ¡¡! O is an origin such that OP1 ´ z1 , OP2 ´ z2 ¡¡! and OP3 ´ z3 . Suppose arg(z2 ¡ z1 ) = ®. a Show that arg(z3 ¡ z2 ) = ® ¡

P2

2¼ 3 .

b Find the modulus and argument of

P1

z2 ¡ z1 . z3 ¡ z2

P3

16 Two of the zeros of P (z) = z 3 + az 2 + bz + c where a, b, c 2 R , are z = ¡3 and z = 3 + i. Find the values of a, b and c.

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17 State the five fifth roots of unity and hence solve: a (2z ¡ 1)5 = 32 b z 5 + 5z 4 + 10z 3 + 10z 2 + 5z = 0

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(z + 1)5 = (z ¡ 1)5

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Chapter

16

Lines and planes in space Contents:

Lines in 2-D and 3-D Applications of a line in a plane Relationship between lines Planes and distances Angles in space The intersection of two or more planes

A B C D E F

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Review set 16A Review set 16B Review set 16C Review set 16D

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LINES AND PLANES IN SPACE (Chapter 16)

INTRODUCTION ³ ´ 1 0

Suppose the vector

represents a displacement of 1 km due East and

³ ´ 0 1

represents a displacement of 1 km due North.

The diagram shows the path of a yacht relative to a yacht club which is situated at (0, 0). At 12:00 noon the yacht is at the point A(2, 20).

y 20

12:00 noon A 1:00 pm 4

(-3 )

2:00 pm

The yacht is travelling in the ³ ´ 4 direction ¡3 with a constant

10

speed of 5 km h¡1 . ¯³ ´¯ ¯ 4 ¯ Since ¯ ¡3 ¯ = 5 we can see that ³ ´ 4 gives both the direction and ¡3

sea x

speed of travel. ³ ´ 4 So, ¡3 is called the velocity

5

10

15

land

vector of the yacht. In order to define the position of the yacht at any time t hours after 12 noon, we can use the parametric equations x = 2 + 4t and y = 20 ¡ 3t where t is called the parameter. Note: If t = 0, x = 2 and y = 20, so the yacht is at (2, 20). If t = 1, x = 6 and y = 17, so the yacht is at (6, 17). If t = 2, x = 10 and y = 14, so the yacht is at (10, 14). We can find a vector equation for the yacht’s path as follows: Suppose the yacht is at R(x, y) at time t hours after 12:00 noon. ¡! ¡! ¡! ) OR = OA + AR ³ ´ ³ ´ 2 4 for t > 0 ) r = 20 + t ¡3 ³ ´ ³ ´ ³ ´ x 2 4 = 20 + t ¡3 ) y

y

A(2, 20)

R(x,¡y)

r

which is the vector equation of the yacht’s path. Notice how the parametric equations are easily found from the vector equation:

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then x = 2 + 4t and y = 20 ¡ 3t.

25

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4 ¡3

5

³ +t

100

´

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2 20

75

³ =

5

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x y

25

³ ´ If

(-43 )

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We can also find the Cartesian equation of the yacht’s path: x¡2 : 4 µ ¶ x¡2 Substituting into y = 20 ¡ 3t gives y = 20 ¡ 3 4 As x = 2 + 4t, 4t = x ¡ 2 and so t =

) 4y = 80 ¡ 3(x ¡ 2) ) 4y = 80 ¡ 3x + 6 ) 3x + 4y = 86 where x > 2 because t > 0.

A

LINES IN 2-D AND 3-D

In both 2-D and 3-D geometry we can determine the equation of a line by its direction and a fixed point through which it passes. Suppose a line passes through a fixed point A such ¡! that OA = a, and its direction is given by vector b (i.e., the line is parallel to b). ¡! Let any point R be on the line so that OR = r. ¡! ¡! ¡! By vector addition, OR = OA + AR. ¡! ¡! Since AR k b, AR = tb for some t 2 R ) r = a + tb r = a + tb, t 2 R

So,

A (fixed point) b a

R (any point)

O (origin)

line

is the vector equation of the line.

LINES IN 2-D ²

In 2-D we are dealing with a line in a plane. ³ ´ ³ ´ ³ ´ x a1 b1 = + t is the vector equation of the line y a2 b2 where R(x, y) is any point on the line A(a1 , a2 ) is the known (fixed) point on the line ³ ´ b b = b1 is the direction vector of the line. 2 µ ¶ ³ ´ b2 rise b1 also enables us to calculate the slope of the line, m = Note: b = b 2 b1 run ¾ x = a1 + b1 t , t 2 R , are the parametric equations of the line y = a2 + b2 t where t is called the parameter. With these equations each point on the line corresponds to exactly one value of t, so every point has a one-to-one correspondence with a real number.

²

²

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We can convert these equations into Cartesian form by equating t values. y ¡ a2 x ¡ a1 = we obtain b2 x ¡ b1 y = b2 a1 ¡ b1 a2 Using t = b1 b2 which is the Cartesian equation of the line.

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Example 1 Find a the vector

b the parametric c

the Cartesian equation ³ ´ 3 of the line passing through the point (1, 5) with direction 2 . a

b

³ ´ ¡! ³ 1 ´ 3 a = OA = 5 and b = 2 ³ ´ ³ ´ ³ ´ x 1 3 = + t , t2R But r = a + tb ) y 5 2 From a, x = 1 + 3t and y = 5 + 2t, t 2 R

R

x¡1 y¡5 Now t = = 3 2 ) 2x ¡ 2 = 3y ¡ 15 ) 2x ¡ 3y = ¡13 fgeneral formg

c

( 32)

A a r O

Example 2 A particle at P(x(t), y(t)) moves such that x(t) = 2 ¡ 3t and y(t) = 2t + 4, t > 0. The distance units are metres and t is in seconds. a Find the initial position of P. b Illustrate the motion showing points where t = 0, 1, 2 and 3. c Find the speed of P. ) the initial position of P is (2, 4)

a

x(0) = 2, y(0) = 4

b

x(1) = ¡1, y(1) = 6 x(2) = ¡4, y(2) = 8 x(3) = ¡7, y(3) = 10

y

t=3

10 t=2 t=1

c

~`2`X`+`\3`X

2

Every second P moves with x-step ¡3 and y-step 2, p which is a distance of 13 m. ) the speed is constant and p is 13 m s¡1 .

5

t=0

3 -10

x -5

5

EXERCISE 16A.1 1 Find i

ii

the vector equation

the parametric equations of the line: ³ ´ 1 a passing through (3, ¡4) with direction 4 ³

b passing through (5, 2) with direction

¡8 2

´

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c cutting the x-axis at ¡6 with direction 3i + 7j d with direction ¡2i + j that passes through (¡1, 11).

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2 Find the parametric equations of the line passing through (¡1, 4) with direction vector ³ ´ 2 and parameter ¸. Find the points on the line when ¸ = 0, 1, 3, ¡1, ¡4. ¡1 a Does (3, ¡2) lie on the line with parametric equations x = t + 2, y = 1 ¡ 3t? Does (0, 6) lie on this line? b (k, 4) lies on the line with parametric equations x = 1 ¡ 2t, y = 1 + t. Find k.

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4 A particle at P(x(t), y(t)) moves such that x(t) = 1 + 2t and y(t) = 2 ¡ 5t, t > 0. The distances are in centimetres and t is in seconds. a Find the initial position of P. b Illustrate the initial part of the motion of P where t = 0, 1, 2, 3. c Find the speed of P.

LINES IN 3-D ²

In 3-D we are dealing with a line in space. Ã ! Ã ! Ã ! x y z

²

=

a1 a2 a3

b1 b2 b3

+¸

where

is the vector equation of the line R(x, y, z) is any point on the line A(a1 , a2 ; a3 ) is the known (fixed) point on the line Ã ! b1 b2 b3

b=

is the direction vector of the line.

We do not talk about the slope of a line in 3-D. We describe its direction only by its direction vector. 9 x = a1 + ¸b1 = y = a2 + ¸b2 are the parametric equations of the line ; z = a3 + ¸b3 where ¸ 2 R is called the parameter.

Note:

²

Every point on the line corresponds to exactly one value of ¸. x ¡ a1 y ¡ a2 z ¡ a3 = = (= ¸) are the Cartesian equations of the line. b1 b2 b3

²

Example 3 Find the vector equation and the parametric equations of the line through (1, ¡2, 3) in the direction 4i + 5j ¡ 6k. Ã ! x y z

The vector equation is

Ã =

1 ¡2 3

!

Ã +¸

4 5 ¡6

! , ¸2R.

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The parametric equations are: x = 1 + 4¸, y = ¡2 + 5¸, z = 3 ¡ 6¸, ¸ 2 R .

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Example 4 Find the parametric equations of the line through A(2, ¡1, 4) and B(¡1, 0, 2). ¡! ¡! We require a direction vector for the line, either AB or BA. ! Ã ! Ã ¡1 ¡ 2 ¡3 ¡! 1 AB = 0 ¡ ¡1 = 2¡4

¡2

Using the point A, the equations are: x = 2 ¡ 3¸, y = ¡1 + ¸, z = 4 ¡ 2¸, ¸ 2 R . Note: Using the point B, the equations are: x = ¡1 ¡ 3¹, y = ¹, z = 2 ¡ 2¹, ¹ 2 R . These sets of equations are actually equivalent, and generate the same set of points. They are related by ¹ = ¸ ¡ 1.

EXERCISE 16A.2 1 Find the vector equation of the line: Ã ! 2 1 3

a parallel to

and through the point (1, 3, ¡7)

b through (0, 1, 2) and with direction vector i + j ¡ 2k c parallel to the X-axis and through the point (¡2, 2, 1). 2 Find the parametric equations of the line: Ã ! ¡1 2 6

a parallel to

and through the point (5, 2, ¡1)

b parallel to 2i ¡ j + 3k and through the point (0, 2, ¡1) c perpendicular to the XOY plane and through (3, 2, ¡1). 3 Find the parametric equations of the line through: a A(1, 2, 1) and B(¡1, 3, 2) b C(0, 1, 3) and D(3, 1, ¡1) c E(1, 2, 5) and F(1, ¡1, 5) d G(0, 1, ¡1) and H(5, ¡1, 3) 4 Find the coordinates of the point where the line with parametric equations x = 1 ¡ ¸, y = 3 + ¸ and z = 3 ¡ 2¸ meets: a the XOY plane b the Y OZ plane c the XOZ plane. 5 Find points on the line pwith parametric equations x = 2 ¡ ¸, y = 3 + 2¸ and z = 1 + ¸ which are 5 3 units from the point (1, 0, ¡2). 6 The perpendicular from a point to a line minimises the distance from the point to that line. Use quadratic theory to find the coordinates of the foot of the perpendicular: a from (1, 1, 2) to the line with equations x = 1 + ¸, y = 2 ¡ ¸, z = 3 + ¸ Ã ! Ã ! Ã ! x y z

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b from (2, 1, 3) to the line with vector equation

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THE ANGLE BETWEEN TWO LINES (2-D and 3-D) In Chapter 14 we saw that the angle between two vectors is measured in the range 0o 6 µ 6 180o . We used the b1 ² b2 formula cos µ = . j b1 j j b2 j In the case of lines which continue infinitely in both directions, we agree to talk about the acute angle between them. We therefore use the formula

q

bx

bx

bz

bz lz

j b1 ² b2 j cos µ = j b1 j j b2 j

q lx

where b1 and b2 are the direction vectors of the given lines l1 and l2 respectively.

Example 5 Find the angle between the lines l1 : x = 2 ¡ 3t, y = ¡1 + t and l2 : x = 1 + 2s, y = ¡4 + 3s ³ ´ ³ ´ ¡3 2 b2 = 3 b1 = 1 j¡6 + 3 j cos µ = p p 10 13

)

) cos µ ¼ 0:2631 and so µ ¼ 74:7o

(1:30 radians)

Example 6 Find the angle between the lines l1 : x = 4 ¡ 3¸, y = ¡1 + ¸, z = 4 ¡ 2¸ 1¡x 2¡z and l2 : =y= 3 2 Ã b1 =

¡3 1 ¡2

!

Ã and b2 =

¡3 1 ¡2

!

j9 + 1 + 4j cos µ = p p 14 14

)

) cos µ = 1 and so µ = 0o In fact these lines are coincident, i.e., l1 and l2 are the same line.

EXERCISE 16A.3

³

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1 Find the angle between the lines: l1 passing through (¡6, 3) parallel to l2 cutting the y-axis at (0, 8) with direction 5i + 4j.

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l1 : x = ¡4 + 12t, y = 3 + 5t and l2 : x = 3s, y = ¡6 ¡ 4s

2 Find the angle between the lines:

x = 2 + 5p, y = 19 ¡ 2p and x = 3 + 4r, y = 7 + 10r are perpendicular.

3 Show that the lines:

x¡8 9¡y z ¡ 10 = = 3 16 7

4 Find the angle between the lines:

and

x = 15 + 3¹, y = 29 + 8¹, z = 5 ¡ 5¹

B APPLICATIONS OF A LINE IN A PLANE THE VELOCITY VECTOR OF A MOVING OBJECT ³

In Example 2 we considered a particle which moves ³ ´ ¡3 is called the velocity vector of the particle. 2

¡3 2

´

every second.

¯³ ´¯ p p ¯ ¡3 ¯ Since ¯ 2 ¯ = (¡3)2 + 22 = 13,

³ ´ p ¡3 , and the velocity of the particle is 13 metres per second in the direction 2 p the speed of the particle is 13 metres per second. In general, ³ ´ a b

if

is the velocity vector of a moving object, then it is travelling ¯³ ´¯ p ³ ´ ¯ a ¯ a at a speed of ¯ b ¯ = a2 + b2 in the direction b .

Example 7 ³ ´ x y

³ ´ =

7 5

³ +t

6 ¡8

´ is the vector equation of the path of an object.

t is the time in seconds, t > 0. The distance units are metres. Find the: a object’s initial position b velocity vector of the object c object’s speed. ³ ´ x y

³ ´ 7 5

At t = 0,

b

The velocity vector is

c

¯³ ´¯ p p ¯ 6 ¯ The speed is ¯ ¡8 ¯ = 36 + 64 = 100 = 10 m s¡1 .

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) the object is at (7, 5).

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CONSTANT VELOCITY PROBLEMS Suppose an object moves with constant velocity b. If the object is initially at A (when time t = 0) and at time t it is at R, then ¡! AR = tb fdistance = time £ speedg ¡! ¡! Now r = OA + AR ) r = a + tb

b

A

R

a r O

Thus if a body has initial position vector a, and moves with constant velocity b, its position at time t is given by r = a + tb for t > 0.

Example 8 An object is initially at (5, 10) and moves with velocity vector 3i ¡ j. Find: a the position of the object at any time t where t is in minutes b the position at t = 3 c the time when the object is due east of (0, 0). a

y (5, 10)

b=

r = a + tb ³ ´ ³ ´ ³ ´ x 5 3 ) = + t , t2R y 10 ¡1 ¶ ³ ´ µ 5 + 3t x ) = 10 ¡ t y

³ ´ 3 ¡1

P

a

r

) P is at (5 + 3t, 10 ¡ t)

x

b

At t = 3, 5 + 3t = 14 and 10 ¡ t = 7 ) it is at (14, 7).

c

When the object is due east of (0, 0), y must be zero ) 10 ¡ t = 0 ) t = 10 The object is due east of (0, 0) after 10 minutes.

THE CLOSEST DISTANCE FROM A POINT TO A LINE A ship sails through point A in the direction b and continues past a port P. At what time will the ship R be closest to the port? b

The ship is closest when PR is perpendicular to AR, ¡ ! ) PR ² b = 0 fthe scalar product is zerog

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Example 9

sea

If distances are measured in kilometres and a ³ ´ 3 ship R moves in the direction 4 at a speed

³ ´ R(x,¡y) 3 4

¡1

of 10 km h , find: a an expression for the position of the ship in terms of t where t is the number of hours after leaving port A b the time when it is closest to port P(10, 2).

A (-8,¡3) O (0,¡0)

P (10,¡2)

land

a

¯³ ´¯ p ¯ 3 ¯ ¯ 4 ¯ = 32 + 42 = 5 ) since the speed is 10 km h¡1 , the ship’s ³ ´ ³ ´ 3 6 velocity vector must be 2 4 = 8 . ¡! ¡! ¡! Now OR = OA + AR ³ ´ ³ ´ ³ ´ x ¡8 6 ) = + t y 3 8 ) R is at (¡8 + 6t, 3 + 8t)

b

¡ ! ³3´ The ship is closest to P when PR ? 4

¡ ! ³3´ PR ² 4 = 0 ¶ ³ ´ µ ¡8 + 6t ¡ 10 3 ² 4 =0 3 + 8t ¡ 2 )

) )

3(6t ¡ 18) + 4(1 + 8t) = 0 ) 18t ¡ 54 + 4 + 32t = 0 ) 50t ¡ 50 = 0 ) t=1 So, the ship is closest to port P 1 hour after leaving A.

EXERCISE 16B.1 1 Each of the following vector equations represents the path of a moving object. t is measured in seconds and t > 0. Distances are measured in metres. In each case find the: i

initial position ii velocity vector iii speed of the object. ³ ´ ³ ´ ³ ´ ³ ´ ³ ´ ³ ´ ³ ´ ³ ´ ³ ´ x ¡4 12 x 0 3 x ¡2 ¡6 = + t b = + t c = ¡7 + t ¡4 a y 3 5 y ¡6 ¡4 y 2 Find the velocity vector of a speed boat moving parallel to: ³ ´ ³ ´ 4 24 ¡1 b with a speed of 150 km h with a speed of 12:5 km h¡1 a ¡3 7 c

2i + j with a speed of 50 km h¡1

d

¡3i + 4j with a speed of 100 km h¡1 .

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3 Yacht A moves according to x(t) = 4 + t, y(t) = 5 ¡ 2t where the distance units are kilometres and the time units are hours. Yacht B moves according to x(t) = 1 + 2t, y(t) = ¡8 + t, t > 0:

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a c d e

Find the initial position of each yacht. b Find the velocity vector of each yacht. Show that the speed of each yacht is constant and state the speeds. If they start at 6:00 am, find the time when the yachts are closest to each other. Prove that the paths of the yachts are at right angles to each other. ³ ´ 3 4 Submarine P is at (¡5, 4) and fires a torpedo with velocity vector ¡1 at 1:34 pm. ³ ´ ¡4 Submarine Q is at (15, 7) and a minutes later fires a torpedo in the direction ¡3 . Distances are measured in kilometres and time is in minutes. a Show that the position of P’s torpedo can be written as P(x1 (t), y1 (t)) where x1 (t) = ¡5 + 3t and y1 (t) = 4 ¡ t. b What is the speed of P’s torpedo? c Show that the position of Q’s torpedo can be written in the form x2 (t) = 15 ¡ 4(t ¡ a), y2 (t) = 7 ¡ 3(t ¡ a): d Q’s torpedo is successful in knocking out P’s torpedo. At what time did Q fire its torpedo and at what time did the explosion occur? ³ ´ 1 5 Let 0 represent a 1 km displacement due east y ³ ´ 100 0 and 1 represent a 1 km displacement due north.

The control tower of an airport is at (0, 0). Aircraft within 100 km of (0, 0) will become visible on the radar screen at the control tower. At 12:00 noon an aircraft is 200¡km east and 100¡km north of the control tower. It is flying parallel to the ³ ´ p ¡3 vector b = ¡1 with a speed of 40 10 km h¡1 .

x -100

100 -100

a Write down the velocity vector of the aircraft. b Write a vector equation for the path of the aircraft using t to represent the time in hours that have elapsed since 12:00 noon. c Find the position of the aircraft at 1:00 pm. d Show that the aircraft first becomes visible on the radar screen at 1:00 pm. e Find the time when the aircraft is closest to the control tower and find the distance between the aircraft and the control tower at this time. f At what time will the aircraft disappear from the radar screen?

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6 Boat A’s position is given by x(t) = 3 ¡ t, y(t) = 2t ¡ 4 where the distance units are kilometres and the time units are hours. Boat B’s position is given by x(t) = 4 ¡ 3t, y(t) = 3 ¡ 2t. a Find the initial position of each boat. b Find the velocity vector of each boat. c What is the angle between the paths of the boats? d At what time are the boats closest to each other?

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GEOMETRIC APPLICATIONS OF r = a + tb Vector equations of two intersecting lines can be solved simultaneously to find the point where the lines meet.

Example 10 ³ ´

³ ´ ³ ´ ¡2 3 = 1 +s 2 and ³ ´ ³ ´ ³ ´ x 15 ¡4 = + t , where s and t are scalars. line 2 has vector equation y 5 1 x y

Line 1 has vector equation

Use vector methods to find where the two lines meet. ³ The lines meet where

¡2 1

´

³ ´ 3 2

+s

) ¡2 + 3s = 15 ¡ 4t ) 3s + 4t = 17 ...... (1) 3s + 4t = 17 8s ¡ 4t = 16 )

11s

³ =

15 5

´

³ +t

¡4 1

´

and 1 + 2s = 5 + t and 2s ¡ t = 4 ...... (2)

f(2) is multiplied by 4g

= 33

So, s = 3 and in (2): 2(3) ¡ t = 4 so t = 2 ³ ´ ³ ´ ³ ´ ³ ´ x ¡2 3 7 Using line 1, = + 3 = 7 y 1 2 ³ ´ ³ ´ ³ ´ ³ ´ x 15 ¡4 7 = + 2 = 7 Checking in line 2, y 5 1

) they meet at (7, 7)

EXERCISE 16B.2 1 The triangle formed by the three lines is ABC. ³ ´ ³ ´ ³ ´ ³ ´ ³ ´ ³ ´ x ¡1 3 x 0 1 Line 1 (AB) is = + r , line 2 (AC) is = + s y 6 ¡2 y 2 1 ³ ´ ³ ´ ³ ´ x 10 ¡2 = ¡3 + t 3 where r, s and t are scalars. and line 3 (BC) is y a b c d

Draw the three lines accurately on a grid. Hence, find the coordinates of A, B and C. Prove that ¢ABC is isosceles. Use vector methods to check your answers to b.

2 A parallelogram is defined by four lines as follows: ³ ´ ³ ´ ³ ´ ³ ´ ³ ´ ³ ´ x ¡4 7 x ¡4 1 Line 1 (AB) is = + r , line 2 (AD) is = + s , y 6 3 y 6 2 ³ ´ ³ ´ ³ ´ ³ ´ ³ ´ ³ ´ x 22 ¡7 x 22 ¡1 = + t , line 4 (CB) is = + u , line 3 (CD) is y 25 ¡3 y 25 ¡2

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where r, s, t and u are scalars.

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a Draw an accurate sketch of the four lines and the parallelogram formed by them. Label the vertices. b From your diagram find the coordinates of A, B, C and D. c Use vector methods to confirm your answers to b. 3 An isosceles triangle ABC is formed by these lines: ³ ´ ³ ´ ³ ´ ³ ´ ³ ´ ³ ´ x 0 2 x 8 ¡1 Line 1 (AB) is = + r , line 2 (BC) is = + s and y 2 1 y 6 ¡2 ³ ´ ³ ´ ³ ´ x 0 1 = + t where r, s and t are scalars. line 3 (AC) is y 5 ¡1 a Use vector methods to find the coordinates of A, B and C. b Which two sides of the triangle are equal in length? Find their lengths. ³ ´

³ ´ ³ ´ ³ ´ ³ ´ 14 x 3 17 + r 10 , line QR is = + s and y ¡1 ¡9 ³ ´ ³ ´ ³ ´ x 0 5 = + t where r, s and t are scalars. line PR is y 18 ¡7 x y

4 Line QP is

³

=

3 ¡1

´

Triangle PQR is formed by these lines. a Use vector methods to find the coordinates of P, Q and R. ¡! ¡! ¡ ¡ ! ! b Find vectors PQ and PR and evaluate PQ ² PR: c Hence, find the size of ]QPR. d Find the area of ¢PQR. 5 Quadrilateral ABCD is formed by these lines: ³ ´ ³ ´ ³ ´ ³ ´ ³ ´ ³ ´ x 2 4 x 18 ¡8 Line 1 (AB) is = + r , line 2 (BC) is = + s , y 5 1 y 9 32 ³ ´ ³ ´ ³ ´ ³ ´ ³ ´ ³ ´ x 14 ¡8 x 3 ¡3 = + t and line 4 (AD) is = + u line 3 (CD) is y 25 ¡2 y 1 12 where r, s, t and u are scalars. a Use vector methods to find the coordinates of A, B, C and D. ¡! ¡! b Write down vectors AC and DB and hence find: ¡! ¡! ¡! ¡! i j AC j ii j DB j iii AC ² DB c What do the answers to b tell you about quadrilateral ABCD?

C

RELATIONSHIP BETWEEN LINES

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the same line infinitely many solutions

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LINE CLASSIFICATION IN 2 DIMENSIONS

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LINE CLASSIFICATION IN 3 DIMENSIONS I I

the lines are coplanar, or lie in the same plane: ² intersecting ² parallel

² coincident

the lines are not coplanar and are hence skew Skew lines are any lines which are neither parallel nor intersecting.

² If the lines are parallel, the angle between them is 0o .

q

² If the lines are intersecting, the angle between them is µ, as shown.

point of intersection

² If the lines are skew, there is still an angle that one line makes with the other. If we translate one line to intersect the other, the angle between the original lines is defined as the angle between the intersecting lines, i.e., angle µ.

line 1

line 1 translated q line 2

meet lines 1 and 2 are skew

Example 11 Line 1 has equations x = ¡1 + 2s, y = 1 ¡ 2s and z = 1 + 4s. Line 2 has equations x = 1 ¡ t, y = t and z = 3 ¡ 2t. Show that the lines are parallel. Ã ! Ã ! Ã ! Ã ! x y z

Line 1 is

¡1 1 1

=

2 ¡2 4

+s

with direction vector Ã ! ¡1 1 ¡2

Likewise, line 2 has direction vector Ã ! Ã ! 2 ¡2 4

Since

¡1 1 ¡2

= ¡2

2 ¡2 4

.

.

, the lines are parallel.

fIf a = kb for some scalar k, then a k b.g

Example 12

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x = ¡1 + 2s, y = 1 ¡ 2s and z = 1 + 4s. x = 1 ¡ t, y = t and z = 3 ¡ 2t. x = 1 + 2u, y = ¡1 ¡ u, z = 4 + 3u. and line 3 intersect and find the angle between them. and line 3 are skew.

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Line 1 has equations Line 2 has equations Line 3 has equations a Show that line 2 b Show that line 1

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a

Equating x, y and z values in lines 2 and 3 gives and 3 ¡ 2t = 4 + 3u 1 ¡ t = 1 + 2u t = ¡1 ¡ u ) t = ¡2u, ) t = ¡1 ¡ u, and 3u + 2t = ¡1 ..... (1) {z } | Solving these we get ¡2u = ¡1 ¡ u ) ¡u = ¡1 ) u = 1 and so t = ¡2 Checking in (1): 3u + 2t = 3(1) + 2(¡2) = 3 ¡ 4 = ¡1 X ) u = 1, t = ¡2 satisfies all three equations, a common solution. Using u = 1, lines 2 and 3 meet at (1 + 2(1), ¡1 ¡ (1), 4 + 3(1)) i.e., (3, ¡2, 7). Ã ! Ã ! ¡1 1 ¡2

Direction vectors for lines 2 and 3 are a = respectively. Now cos µ =

and b =

2 ¡1 3

ja ² bj j¡2 ¡ 1 ¡ 6j p , where µ is the acute = p jajjbj 1+1+4 4+1+9

9 angle between a and b ) cos µ = p 84

and so µ ¼ 10:89o

) the angle between lines 2 and 3 is about 10:9o . b

Equating x, y and z values in lines 1 and 3 gives and 1 + 4s = 4 + 3u ¡1 + 2s = 1 + 2u 1 ¡ 2s = ¡1 ¡ u and 4s ¡ 3u = 3 ..... (1) ) 2s ¡ 2u = 2, ) ¡2s + u = ¡2, {z } | Solving these we get 2s ¡ 2u = 2 ) ¡2s + u = ¡2 ) ¡u = 0 fadding themg ) u = 0 and so 2s = 2 i.e., s = 1 Checking in (1), 4s ¡ 3u = 4(1) ¡ 3(0) = 4 6= 3 So, there is no simultaneous solution to all 3 equations. ) the lines cannot meet, and as they are not parallel they must be skew.

Example 13

½

Discuss the solutions to

3x ¡ y = 2 6x ¡ 2y = k

for k 2 R .

Give a geometric interpretation of the solutions. · In augmented matrix form If k ¡ 4 = 0,

¡1 ¡2

3 6

2 k

¸

· »

3 0

¡1 0

2 k¡4

¸ R2 ¡ 2R1

the system is consistent, and the lines 3x ¡ y = 2 and 6x ¡ 2y = 4 are coincident.

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Let x = t, so y = ¡2 + 3t. There are infinitely many solutions of the form x = t, y = ¡2 + 3t, t 2 R .

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³ ´ 1 3

This line has direction vector If k ¡ 4 6= 0

or slope 3 and passes through the point (0, ¡2).

i.e., k 6= 4 the system is inconsistent. In this case the lines 3x ¡ y = 2 and 6x ¡ 2y = k are parallel and have no points of intersection.

PERPENDICULAR AND PARALLEL TESTS (for 2-D AND 3-D) I I

Non-zero vectors v and w are

perpendicular if v ² w = 0 parallel if v = kw for some scalar k

If v ² w = 0 then j v j j w j cos µ = 0 ) cos µ = 0 fv, w non-zerog ) µ = 90o

Proof of perpendicular case:

EXERCISE 16C 1 Classify the following line pairs as either parallel, intersecting or skew, and in each case find the measure of the acute angle between them: a x = 1 + 2t, y = 2 ¡ t, z = 3 + t and x = ¡2 + 3s, y = 3 ¡ s, z = 1 + 2s b x = ¡1 + 2¸, y = 2 ¡ 12¸, z = 4 + 12¸ and x = 4¹ ¡ 3, y = 3¹ + 2, z = ¡¹ ¡ 1 c x = 6t, y = 3 + 8t, z = ¡1 + 2t and x = 2 + 3s, y = 4s, z = 1 + s d x = 2 ¡ y = z + 2 and x = 1 + 3s, y = ¡2 ¡ 2s, z = 2s + 12 e x = 1 + ¸, y = 2 ¡ ¸, z = 3 + 2¸ and x = 2 + 3¹, y = 3 ¡ 2¹, z = ¹ ¡ 5 f x = 1 ¡ 2t, y = 8 + t, z = 5 and x = 2 + 4s, y = ¡1 ¡ 2s, z = 3 2 Consider the two lines whose equations are 3x ¡ y = 8 and 6x ¡ 2y = k where k is some real number. Discuss the nature of the intersection of these lines for different values of k. 3 Discuss for the different values of a, the geometric solutions of the equations 4x + 8y = 1 and 2x ¡ ay = 11.

SHORTEST DISTANCE FROM A POINT TO A LINE (2-D AND 3-D)

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The coordinates of any point P on a line can be expressed in terms of the parameter t. ¡ ! We can hence find the vector AP in terms of t, where A is a point which is not on the line. ¡ ! The shortest distance d occurs when AP is perpendicular to b, ¡ ! so we find t for which AP ² b = 0. ¡! The shortest distance d = j AP j for this value of t.

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Example 14 Find the shortest distance from P(¡1, 2, 3) to the line

x¡1 y+4 = = z ¡ 3. 2 3

The equation of the line in parametric form is x = 1 + 2¸, y = ¡4 + 3¸, z = 3 + ¸ ) any point A on this line has coordinates (1 + 2¸, ¡4 + 3¸, 3 + ¸) ! Ã ! Ã 2 + 2¸ 2 ¡ ! and the direction vector of the line is b = 3 ) PA = ¡6 + 3¸ 1

¸

¡ ! Now for the shortest distance, PA ² b = 0 fperpendicular distanceg ) 4 + 4¸ ¡ 18 + 9¸ + ¸ = 0 ) 14¸ = 14 and so ¸ = 1 Ã ! 4 p ¡ ! ¡ ! and d = j PA j = 26 units Thus PA = ¡3

A

P(-1, 2, 3)

1

4 Find the shortest distance from the point (2, ¡3) to the line 3x ¡ y = 4: 5 Find the shortest distance from the point (3, 0, ¡1) to the line with equation r = 2i ¡ j + 4k + ¸(3i + 2j + k): Ã ! Ã ! 1 ¡1 2

6 Find the shortest distance from (1, 1, 3) to the line r =

+¸

2 3 1

.

THE SHORTEST DISTANCE BETWEEN SKEW LINES (EXTENSION) Example 15 Find the shortest distance between the skew lines x = t, y = 1 ¡ t, z = 2 + t and x = 3 ¡ s, y = ¡1 + 2s, z = 4 ¡ s. Ã The lines have direction vectors v = ¯ ¯ i ¯ so v £ w = ¯ 1 ¯ ¡1

¯

k ¯ ¯ 1 ¯ ¡1 ¯

j ¡1 2

¯ ¯ ¡1 =¯ 2

1 ¡1 1

¯

1 ¯ i ¡1 ¯

!

Ã and w =

¯ ¯ 1 ¡ ¯ ¡1

¯

1 ¯ j ¡1 ¯

¡1 2 ¡1

!

¯ ¯ 1 + ¯ ¡1

Ã

¯

¡1 ¯ k 2 ¯

=

¡1 0 1

!

Let A and B be points on the skew lines. ) A is (t, 1 ¡ t, 2 + t) and B is (3 ¡ s, ¡1 + 2s, 4 ¡ s) where s and t are scalars. ¡! To find the shortest distance between the two skew lines, we need to find j AB j ¡! where AB is parallel to v £ w, i.e., perpendicular to both skew lines. ! Ã ! Ã 3¡s¡t ¡1 ¡! ¡1 + 2s ¡ 1 + t =k 0 for some scalar k. As AB k v £ w,

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4¡s¡2¡t

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) 3 ¡ s ¡ t = ¡k, Thus 2s + t = 2 and ) 2s + t = 2 and

¡2 + 2s + t = 0 and 2 ¡ s ¡ t = k 3 ¡ s ¡ t = ¡2 + s + t 2s + 2t = 5

Solving simultaneously, t = 3 and s = ¡ 12 0 1 1 q ¡! ¡! @ 2 A and j AB j = 14 + 0 + ) AB = 0 ¡ 12

1 4

=

p1 2

units p1 2

) the shortest distance is

units.

7 Find the shortest distance between the skew lines: a x = 1 + 2t, y = ¡t, z = 2 + 3t and x = y = z b x = 1 ¡ t, y = 1 + t, z = 3 ¡ t and x = 2 + s, y = 1 ¡ 2s, z = s 8 Find the shortest distance between the lines given in Exercise 16C question 1. ²

Note:

²

To find the distance between parallel lines, find the distance from a point on one line to the other line. What is the shortest distance between intersecting/coincident lines?

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PLANES AND DISTANCES

To find the equation of a plane, we need to know a point on the plane and also its orientation in space. The orientation of a plane cannot be given by a single parallel vector because infinitely many planes of different orientation are parallel to a single direction vector. We require two non-parallel vectors to define the orientation uniquely. Any point R(x, y, z) on the plane with a known point A(a1 , a2 , a3 ) and two non-parallel Ã ! Ã ! b1 b2 b3

vectors b =

and c =

c1 c2 c3

must satisfy the vector equation

¡! AR = ¸b + ¹c for some scalars ¸ and ¹. ¡! ¡! ) OR ¡ OA = ¸b + ¹c ¡! ¡! ) OR = OA + ¸b + ¹c r = a + ¸b + ¹c is the vector equation of the plane

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where r is the position vector of any point on the plane, a is the position vector of the known point A(a1 , a2 , a3 ) on the plane b and c are any two non-parallel vectors that are parallel to the plane ¸, ¹ 2 R are two independent parameters.

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Another way of defining the direction of a plane is to consider the vector cross product of the two vectors b and c which are parallel to the plane. The vector n = b £ c is called a normal vector to the plane. n is perpendicular to b and c and is hence perpendicular to any vector or line in or parallel to the plane. This is because any vector parallel to the plane can be written in the form ¸b + ¹c. Challenge: Show n ? ¸b + ¹c. Ã ! a b c

Suppose a plane in space has normal vector n = and that it includes the fixed point A(x1 , y1 , z1 ). R(x, y, z) is any other point in the plane. ¡! Now AR is perpendicular to n ¡! ) n ² AR = 0 ! Ã ! Ã a b c

)

x ¡ x1 y ¡ y1 z ¡ z1

²

Ã !

a

n= b c

R(x, y, z) A(xz, yz, zz)

=0

) a(x ¡ x1 ) + b(y ¡ y1 ) + c(z ¡ z1 ) = 0 ) ax + by + cz = ax1 + by1 + cz1

where the RHS is a constant.

¡! n ² AR = 0 is another form of the vector equation of the plane. It could also be written as n ² (r ¡ a) = 0, which implies r ² n = a ² n. Ã ! a b c

If a plane has normal vector n =

and passes through (x1 , y1 , z1 )

then it has equation ax + by + cz = ax1 + by1 + cz1 = d, where d is a constant. This is the Cartesian equation of the plane.

Example 16

Ã !

Find the equation of the plane with normal vector

1 2 3

and containing (¡1, 2, 4).

Ã ! Since n =

1 2 3

and (¡1, 2, 4) lies on the plane, the equation is

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x + 2y + 3z = (¡1) + 2(2) + 3(4) i.e., x + 2y + 3z = 15

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Example 17 Find the equation of the plane through A(¡1, 2, 0), B(3, 1, 1) and C(1, 0, 3): a in vector form b in Cartesian form Ã ! Ã ! 4 2 ¡! ¡! 1 and CB = a AB = ¡1 ¡2

1

¡! ¡! Now AB and CB are two non-parallel vectors both parallel to the plane. Ã ! Ã ! Ã ! Ã ! ) r=

x y z

1 0 3

=

+¸

4 ¡1 1

2 1 ¡2

+¹

, ¸, ¹ 2 R

using C as the known (fixed) point on the plane. Ã ! Ã ! 4 2 ¡! ¡! If n is the normal vector, then n = AB £ AC = ¡1 £ ¡2

b

¯ ¯ ¡1 ) n = ¯ ¡2

¯

1¯ 3¯

¯ ¯4 i ¡ ¯2

¯

1¯ 3¯

¯ ¯4 j + ¯2

Ã

¯

¡1 ¯ ¡2 ¯

k=

1 ¡1 ¡10 ¡6

!

Ã =¡

3 1 10 6

!

Thus the plane has equation x + 10y + 6z = (¡1) + 10(2) + 6(0) fusing Ag ) x + 10y + 6z = 19 Note: Check that all 3 points satisfy this equation.

Example 18 Find the parametric equations of the line through A(¡1, 2, 3) and B(2, 0, ¡3) and hence find where this line meets the plane with equation x ¡ 2y + 3z = 26: Ã ! 3 ¡! so line AB has parametric equations AB = ¡2 ¡6 x = ¡1 + 3t, y = 2 ¡ 2t, z = 3 ¡ 6t ..... (¤) This line meets the plane x ¡ 2y + 3z = 26 where ¡1 + 3t ¡ 2(2 ¡ 2t) + 3(3 ¡ 6t) = 26 ) 4 ¡ 11t = 26 ) ¡11t = 22 and so t = ¡2 ) it meets the plane at (¡7, 6, 15) fsubstituting t = ¡2 into ¤g

Example 19 Find the coordinates of the foot of the normal from A(2, ¡1, 3) to the plane x ¡ y + 2z = 27. Hence find the shortest distance from A to the plane. Ã ! A

x ¡ y + 2z = 27 has normal vector

n

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) the parametric equations of AN are x = 2 + t, y = ¡1 ¡ t, z = 3 + 2t

N

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Mathematics for the International Students: IB Dipolma HL Core

Mathematics for the International Students: IB Dipolma HL Core, Second Edition

Mathematics HL (Core) 2nd. edtion

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